The notes of this book originate from three series of lectures given at the Centre de Recerca Matemàtica (CRM) in Barcelona. The first one is dedicated to the study of periodic solutions of autonomous differential systems in R<sup>n via the Averaging Theory and was delivered by Jaume Llibre. The second one, given by Richard Moeckel, focusses on methods for studying Central Configurations. The last one, by Carles Simó, describes the main mechanisms leading to a fairly global description of the dynamics in conservative systems.

The book is directed towards graduate students and researchers interested in dynamical systems, in particular in the conservative case, and aims at facilitating the understanding of dynamics of specific models. The results presented and the tools introduced in this book include a large range of applications.

Advanced Courses in Mathematics CRM Barcelona

Jaume Llibre Richard Moeckel Carles Simó

Central Conﬁgurations, Periodic Orbits, and Hamiltonian Systems

Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Enric Ventura

More information about this series at http://www.springer.com/series/5038

Jaume Llibre • Richard Moeckel • Carles Simó

Central Configurations, Periodic Orbits, and Hamiltonian Systems Editors for this volume: Montserrat Corbera, Universitat de Vic Josep Maria Cors, Universitat Politècnica de Catalunya Enrique Ponce, Universidad de Sevilla

Jaume Llibre Departament de Matemàtiques Universitat Autònoma de Barcelona Barcelona, Spain

Richard Moeckel School of Mathematics University of Minnesota Minneapolis, MN, USA

Carles Simó Dept Matemàtica Aplicada i Anàlisi Universitat de Barcelona Barcelona, Spain

ISSN 2297-0304 ISSN 2297-0312 (electronic) Advanced Courses in Mathematics - CRM Barcelona ISBN 978-3-0348-0932-0 ISBN 978-3-0348-0933-7 (eBook) DOI 10.1007/978-3-0348-0933-7 Library of Congress Control Number: 2015956332 Mathematics Subject Classification (2010): 34C29, 34C25, 37J10, 37J15, 37J40, 37J45, 37M05, 37N05, 47H11, 70F07, 70F10, 70F15 Springer Basel Heidelberg New York Dordrecht London © Springer Basel 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)

Foreword This book collects the notes of lectures given by Jaume Llibre, Richard Moeckel, and Carles Sim´o at Centre de Recerca Matem`atica (CRM) in Bellaterra, Barcelona, from January 27th to 31st, 2014. The activity, in the framework of the Research Program on Central Conﬁgurations, Periodic Orbits and Beyond in Celestial Mechanics, hosted at CRM from January to July 2014, was a joint collaboration with the winter school in dynamical systems Recent Trends in Nonlinear Science (RTNS2014), promoted by the DANCE (Din´amica, Atractores y Nolinealidad: Caos y Estabilidad) Spanish network. The Advanced Course on Central Conﬁgurations, Periodic Orbits and Hamiltonian Systems aimed at training their participants both theoretically and in applications in the ﬁeld of nonlinear science; in this area as in many others, the theoretical and the applications points of view clearly reinforce each other. There were three series of lectures and, accordingly, the material is distributed in three chapters in the book. The ﬁrst series, delivered by Jaume Llibre, was dedicated to the study of periodic solutions of diﬀerential systems in Rn via Averaging Theory. Roughly speaking, in Averaging Theory one replaces a vector ﬁeld by its average (over time or an angular variable) with the goal of obtaining asymptotic approximations to the original system that will be capable of guaranteeing the existence of periodic solutions. The corresponding notes in Chapter 1 start with an introduction of the classical, ﬁrst order averaging theory followed by the main results of the theory for arbitrary order and dimension. The theory is applied next to the study of periodic solutions of some well known diﬀerential equations, like the van der Pol diﬀerential equation, the Li´enard diﬀerential systems, or the Rossler diﬀerential system, among others. Some Hamiltonian systems are also studied. The second series of lectures, given by Richard Moeckel, focused on methods for studying central conﬁgurations, in Chapter 2. A Central Conﬁguration is a special arrangement of point masses interacting by Newton’s law of gravitation, and with the following property: the gravitational acceleration vector produced on each mass by all the others should point toward the center of mass and be proportional to the distance to the center of mass. Central Conﬁgurations play an important role in study of the Newtonian n-body problem. For example, they lead to the only explicit solutions of the equations of motion, they govern the behavior of solutions near collisions, and they inﬂuence the topology of integral manifolds. The

v

vi

Foreword

lectures dealt with questions about the existence and enumeration of various types of Central Conﬁgurations, including algebraic-geometrical approaches to Smale’s Sixth Problem: is the number of Central Conﬁgurations always ﬁnite? Chapter 3 is devoted to the last series of lectures, given by Carles Sim´o. They describe the main mechanisms leading to a fairly global description of the dynamics in conservative systems, either in the continuous version described by a Hamiltonian, or in the discrete version. The Newtonian n-body problem belongs to the general class of Hamiltonian systems. The chapter starts with several simple but paradigmatic examples in the 2D case, from which it is easier to grasp the main underlying ideas, also useful in higher dimension. Next, general theoretical results are presented and applied to diﬀerent problems in Celestial Mechanics, with a rich variety of goals. We would like to express our gratitude to the director and staﬀ of the Centre de Recerca Matem`atica for making possible this activity. Finally, our special thanks to the three lecturers, Jaume Llibre, Richard Moeckel and Carles Sim´ o, for the enthusiasm they showed during the course and for their ﬁne preparation of these notes. It is our hope that with their publication we may contribute to the spreading of the interest of actual and future researchers for the exciting world of dynamical systems.

Montserrat Corbera, Josep M. Cors and Enrique Ponce

Contents 1

The Averaging Theory for Computing Periodic Orbits Jaume Llibre 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Introduction: the classical theory . . . . . . . . . . . . . . . . . 1.2.1 A ﬁrst order averaging method for periodic orbits . . . . 1.2.2 Four applications . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Other ﬁrst order averaging methods for periodic orbits . 1.2.4 Three applications . . . . . . . . . . . . . . . . . . . . . 1.2.5 Another ﬁrst order averaging method for periodic orbits 1.2.6 Proof of Theorem 1.2.1 . . . . . . . . . . . . . . . . . . 1.2.7 Proof of Theorem 1.2.9 . . . . . . . . . . . . . . . . . . 1.2.8 Proof of Theorem 1.2.18 . . . . . . . . . . . . . . . . . . 1.3 Averaging theory for arbitrary order and dimension . . . . . . . 1.3.1 Statement of the main results . . . . . . . . . . . . . . . 1.3.2 Proofs of Theorems 1.3.5 and 1.3.6 . . . . . . . . . . . . 1.3.3 Computing formulae . . . . . . . . . . . . . . . . . . . . 1.3.4 Fifth order averaging of Theorem 1.3.5 . . . . . . . . . . 1.3.5 Fourth order averaging of Theorem 1.3.6 . . . . . . . . . 1.3.6 Appendix: basic results on the Brouwer degree . . . . . 1.4 Three applications of Theorem 1.3.5 . . . . . . . . . . . . . . . 1.4.1 The averaging theory of ﬁrst, second and third order . . 1.4.2 The H´enon–Heiles Hamiltonian . . . . . . . . . . . . . . 1.4.3 Limit cycles of polynomial diﬀerential systems . . . . . 1.4.4 The generalized polynomial diﬀerential Li´enar equation Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Central Conﬁgurations Richard Moeckel 2.1 The n-body problem . . . . . . . . . . . . . . . 2.2 Symmetries and integrals . . . . . . . . . . . . 2.3 Central conﬁgurations and self-similar solutions 2.4 Matrix equations of motion . . . . . . . . . . .

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1 1 2 2 4 17 18 32 35 44 45 46 47 51 58 59 60 63 64 64 66 72 79 99

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105 105 106 108 112

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Contents 2.5 Homographic motions of central conﬁgurations in Rd . . . 2.6 Albouy–Chenciner reduction and relative equilibria in Rd 2.7 Homographic motions in Rd . . . . . . . . . . . . . . . . . 2.8 Central conﬁgurations as critical points . . . . . . . . . . 2.9 Collinear central conﬁgurations . . . . . . . . . . . . . . . 2.10 Morse indices of non-collinear central conﬁgurations . . . 2.11 Morse theory for CC’s and SBC’s . . . . . . . . . . . . . . 2.12 Dziobek conﬁgurations . . . . . . . . . . . . . . . . . . . . 2.13 Convex Dziobek central conﬁgurations . . . . . . . . . . . 2.14 Generic ﬁniteness for Dziobek central conﬁgurations . . . 2.15 Some open problems . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Dynamical Properties of Hamiltonian Systems with Applications to Celestial Dynamics Carles Sim´ o 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Continuous and discrete conservative systems . . . . . . . 3.1.2 Comments on the contents . . . . . . . . . . . . . . . . . 3.2 Low dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The H´enon map . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The standard map . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Return maps: the separatrix map . . . . . . . . . . . . . . 3.3 Some theoretical results, their implementation and practical tools 3.3.1 A preliminary tool: the integration of the ODE, Taylor method and jet transport . . . . . . . . . . . . . . . . . . 3.3.2 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Stability results: KAM theory and related topics . . . . . 3.3.4 Invariant manifolds . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Instability, bounds and detection . . . . . . . . . . . . . . 3.4 Applications to Celestial Mechanics . . . . . . . . . . . . . . . . . 3.4.1 An elementary mission around L1 . . . . . . . . . . . . . 3.4.2 Escape and conﬁnement in the Sitnikov problem . . . . . 3.4.3 Practical conﬁnement around triangular points . . . . . . 3.4.4 Inﬁnitely many choreographies in the three-body problem 3.4.5 Evidences of diﬀusion related to the centre manifold of L3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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116 119 127 129 138 144 146 151 154 157 162 165

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169 169 170 173 173 174 182 184 188

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188 191 193 196 198 201 202 205 209 215 219 227

Chapter 1

The Averaging Theory for Computing Periodic Orbits Jaume Llibre 1.1 Preface The method of averaging is a classical tool allowing us to study the dynamics of the non-linear diﬀerential systems under periodic forcing. The method of averaging has a long history starting with the classical works of Lagrange and Laplace, who provided an intuitive justiﬁcation of the method. The ﬁrst formalization of this theory was done in 1928 by Fatou [34]. Important practical and theoretical contributions to the averaging theory were made in the 1930’s by Bogoliubov– Krylov [8], in 1945 by Bogoliubov [7], and by Bogoliubov–Mitropolsky [9] (english version 1961). For a more modern exposition of the averaging theory, see the book Sanders–Verhulst–Murdock [78]. Every orbit of a diﬀerential system is homeomorphic either to a point, or to a circle, or to a straight line. In the ﬁrst case it is called a singular point or an equilibrium point, and in the second case it is called a periodic orbit. The third case does not have a name. These notes are dedicated to studying analytically the periodic orbits of a given diﬀerential system. We consider diﬀerential systems of the form x˙ = F0 (t, x) + εF1 (t, x) + ε2 R(t, x, ε),

(1.1)

with x in some open subset D of Rn , Fi : R × D → Rn of class C 2 for i = 1, 2, R : R × D × (−ε0 , ε0 ) → Rn of class C 2 with ε0 > 0 small, and with the functions Fi and R being T -periodic in the variable t. Here, the dot denotes derivative with respect to the time t.

© Springer Basel 2015 J. Llibre et al., Central Configurations, Periodic Orbits, and Hamiltonian Systems, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0933-7_1

1

2

Chapter 1. The Averaging Theory for Computing Periodic Orbits

In general, to obtain analytically periodic solutions of a diﬀerential system is a very diﬃcult problem, many times a problem impossible to solve. As we shall see when we can apply the averaging theory, this diﬃcult problem for diﬀerential systems (1.1) is reduced to ﬁnding the zeros of a non-linear function of dimension at most n, i.e., now the problem has the same diﬃculty as the problem of ﬁnding the singular or equilibrium points of a diﬀerential system. An important problem for studying periodic solutions of diﬀerential systems of the form x˙ = F (t, x), or x˙ = F (x), (1.2) using averaging theory is to transform them into systems written in the normal form of the averaging theory, i.e., as a system (1.1). Note that systems (1.2), in general, are not periodic in the independent variable t and do not have any small parameter ε. So, we must ﬁnd changes of variables which allow us to write the diﬀerential systems (1.2) into the form (1.1), where F0 can eventually be zero. The present chapter is divided in three sections. Section 1.2 is dedicated to the averaging theory of ﬁrst order; we present in it three main results for studying the periodic solutions of diﬀerential systems, see Theorems 1.2.1, 1.2.9 and 1.2.18. We develop four applications of Theorems 1.2.1, namely to the van der Pol equation, to the Li´enard diﬀerential system, to study the zero-Hopf bifurcation in Rn , and to a class of Hamiltonian systems. We present three applications of Theorem 1.2.9; in the ﬁrst we study the Hopf bifurcation of the Michelson system, in the second the periodic solutions of a third-order diﬀerential equation, and in the third one we analyze the periodic solutions of the Vallis system which models the “El Ni˜ no” phenomenon. Finally, we do an application of Theorem 1.2.18 to a class of Duﬃng diﬀerential equations. In Section 1.3, the most theoretical one, we present averaging theory for studying periodic solutions of a diﬀerential system in Rn at any order in the small parameter. This theory is developed using weaker assumptions. Finally, in Section 1.4, we present some applications of averaging theory of order higher than one. More concretely, using averaging theory of second order we study periodic solutions of the H´enon–Heiles Hamiltonian, and using averaging theory of third order we study ﬁrst the limit cycles of the quadratic polynomial diﬀerential systems, and of the linear with cubic homogeneous non-linearities polynomial diﬀerential systems; and ﬁnally, we analyze the periodic solutions of the generalized Li´enard polynomial diﬀerential equations.

1.2 Introduction: the classical theory 1.2.1 A ﬁrst order averaging method for periodic orbits We consider the diﬀerential system x˙ = εF (t, x) + ε2 R(t, x, ε),

(1.3)

1.2. Introduction: the classical theory

3

with x ∈ D ⊂ Rn , D a bounded domain, and t ≥ 0. Moreover we assume that F (t, x) and R(t, x, ε) are T -periodic in t. The averaged system associated to the system(1.3) is deﬁned by y˙ = εf 0 (y),

(1.4)

where

1 T F (s, y)ds. (1.5) T 0 The next theorem says under what conditions the singular points of the averaged system (1.4) provide T -periodic orbits for the system (1.3). The proof presented here comes from [85]. f 0 (y) =

Theorem 1.2.1. We consider system (1.3) and assume that the vector functions F , R, Dx F , Dx2 F and Dx R are continuous and bounded by a constant M (independent of ε) in [0, ∞) × D, with −ε0 < ε < ε0 . Moreover, we suppose that F and R are T -periodic in t, with T independent of ε. (i) If p ∈ D is a singular point of the averaged system (1.4) such that det(Dx f 0 (p)) = 0

(1.6)

then, for |ε| > 0 suﬃciently small, there exists a T -periodic solution x(t, ε) of system (1.3) such that x(0, ε) → p as ε → 0. (ii) If the singular point y = p of the averaged system (1.4) has all its eigenvalues with negative real part then, for |ε| > 0 suﬃciently small, the corresponding periodic solution x(t, ε) of system (1.3) is asymptotically stable and, if one of the eigenvalues has positive real part x(t, ε), it is unstable. Theorem 1.2.1 is proved in Subsection 1.2.6. Before its proof we shall present some applications of it in Subsection 1.2.2. For each z ∈ D we denote by x(·, z, ε) the solution of (1.3) with initial condition x(0, z, ε) = z. We consider also the function ζ : D × (−ε0 , ε0 ) → Rn deﬁned by T ζ(z, ε) = εF (t, x(t, z, ε)) + ε2 R(t, x(t, z, ε), ε) dt. (1.7) 0

From (1.3) it follows that, for every z ∈ D, ζ(z, ε) = x(T, z, ε) − x(0, z, ε).

(1.8)

The function ζ can be written in the form ζ(z, ε) = εf 0 (z) + O(ε2 ), 0

(1.9)

where f is given by (1.5). Moreover, under the assumptions of Theorem 1.2.1, the solution x(t, ε), for |ε| suﬃciently small, satisﬁes that zε = x(0, ε) tends to be an isolated zero of ζ(·, ε) when ε → 0. Of course, due to (1.8) the function ζ is a displacement function for system (1.3), and its ﬁxed points are initial conditions for the T -periodic solutions of system (1.3).

4

Chapter 1. The Averaging Theory for Computing Periodic Orbits

1.2.2 Four applications We recall that a limit cycle of a diﬀerential system is a periodic orbit isolated in the set of all periodic orbits of the system. The van der Pol diﬀerential equation Consider the van der Pol diﬀerential equation x ¨ + x = ε(1 − x2 )x, ˙ which can be written as the diﬀerential system x˙ = y, y˙ = −x + ε(1 − x2 )y.

(1.10)

In polar coordinates (r, θ), where x = r cos θ, y = r sin θ, this system becomes r˙ = εr(1 − r2 cos2 θ) sin2 θ, θ˙ = −1 + ε cos θ(1 − r2 cos2 θ) sin θ, or, equivalently, dr = −εr(1 − r2 cos2 θ) sin2 θ + O(ε2 ). dθ Note that the previous diﬀerential system is in the normal form (1.3) for applying the averaging theory described in Theorem 1.2.1 if we take x = r, t = θ, T = 2π and F (t, x) = −r(1 − r2 cos2 θ) sin2 θ. From (1.5) we get that 2π 1 1 r(1 − r2 cos2 θ) sin2 θdθ = r(r2 − 4). f 0 (r) = − 2π 0 8 The unique positive root of f 0 (r) is r = 2. Since (df 0 /dr)(2) = 1, by Theorem 1.2.1 (i), it follows that system (1.10) has, for |ε| = 0 suﬃciently small, a limit cycle bifurcating from the periodic orbit of radius 2 of the unperturbed system (1.10) with ε = 0. Moreover, since (df 0 /dr)(2) = 1 > 0, by Theorem 1.2.1 (ii), this limit cycle is unstable. The Li´enard diﬀerential system The following result is due to Lins–de Melo–Pugh [53]. Here, we provide an easy and shorter proof with respect to the initial proof given by the mentioned authors. Proposition 1.2.2. The Li´enard diﬀerential systems of the form x˙ = y − ε(a1 x + · · · + an xn ), y˙ = −x, with ε suﬃciently small and an = 0 have at most [(n−1)/2] limit cycles bifurcating from the periodic orbits of the linear center x˙ = y, y˙ = −x, and there are examples with exactly [(n − 1)/2] limit cycles; here, [·] denotes the integer part function.

1.2. Introduction: the classical theory

5

Proof. We write the system x˙ = y − ε(a1 x + · · · + an xn ),

y˙ = −x,

in polar coordinates (r, θ), where x = r cos θ, y = r sin θ, and we obtain r˙ = −ε

n

ak rk cosk+1 θ,

k=1

θ˙ = −1 + ε sin θ

n

ak rk−1 cosk θ

k=1

or, equivalently, dr = −ε ak rk cosk+1 θ + O(ε2 ). dθ n

k=1

Again, taking x = r, t = θ, T = 2π and F (t, x) = −

n

ak rk cosk+1 θ, the previous

k=1

diﬀerential system is in the normal form (1.3) for applying the averaging theory described in Theorem 1.2.1. We have that 2π n 1 ε k f (r) = − ak r cosk+1 θ dθ = − 2π 2π 0 0

k=1

where bk =

2π

n

ak bk rk = p(r),

k=1 k odd

cosk+1 θ dθ = 0 if k is odd, and bk = 0 if k is even. Now we

apply Theorem 1.2.1, since the polynomial p(r) has at most [(n − 1)/2] positive roots, and we can choose the coeﬃcients ak with k odd in such a way that p(r) has exactly [(n − 1)/2] simple positive roots; the proposition follows. Zero-Hopf bifurcation in Rn In this example we study a zero-Hopf bifurcation of C 3 diﬀerential systems in Rn with n ≥ 3. These results come from Llibre–Zhang [58]. We assume that these systems have a singularity at the origin, whose linear part has eigenvalues εa ± bi, with b = 0 and εck for k = 3, . . . , n, where ε is a small parameter. Since the eigenvalues of the linearization at the origin when ε = 0 are ±bi = 0 and 0 with multiplicity n − 2, if an inﬁnitesimal periodic orbit bifurcates from the origin when ε = 0, we call such kind of bifurcation a zero-Hopf

6

Chapter 1. The Averaging Theory for Computing Periodic Orbits

bifurcation. Such systems can be written into the form

x˙ = εax − by +

i1 +···+in =2

bi1 ...in xi1 y i2 z3i3 · · · znin + B, i1 +···+in =2 (k) ci1 ...in xi1 y i2 z3i3 · · · znin + Ck , k i1 +···+in =2

y˙ = bx + εay + z˙k = εck zk +

ai1 ...in xi1 y i2 z3i3 · · · znin + A, (1.11) = 3, . . . , n,

(k)

where ai1 ...in , bi1 ...in , ci1 ...in , a, b and ck are real parameters, ab = 0, and A, B and Ck are the Lagrange expression of the error function of third order in the expansion of the functions of the system in Taylor series. Theorem 1.2.3. There exist C 3 systems (1.11) for which l ∈ {0, 1, . . . , 2n−3 } limit cycles bifurcate from the origin at ε = 0, i.e., for ε suﬃciently small the system has exactly l limit cycles in a neighborhood of the origin, and these limit cycles tend to the origin when ε 0. As far as we know, Theorem 1.2.3 was the ﬁrst result proving that the number of limit cycles that can bifurcate in a Hopf bifurcation increases exponentially with the dimension of the space. We recall that a Hopf bifurcation takes place when one or several limit cycles bifurcate from an equilibrium point. From the proof of Theorem 1.2.3 we get immediately the following result. Corollary 1.2.4. There exist quadratic polynomial diﬀerential systems (1.11) (i.e., with A = B = Ck = 0) for which l ∈ {0, 1, . . . , 2n−3 } limit cycles bifurcate from the origin at ε = 0, i.e., for ε suﬃciently small the system has exactly l limit cycles in a neighborhood of the origin and these limit cycles tend to the origin when ε 0. Proof of Theorem 1.2.3. Doing the cylindrical change of coordinates x = r cos θ,

y = r sin θ,

zi = zi ,

i = 3, . . . , n,

(1.12)

in the region r > 0 the system (1.11) becomes r˙ = εar +

(ai1 ...in cos θ + bi1 ...in sin θ)(r cos θ)i1 (r sin θ)i2 z3i3 · · · znin + O(3),

i1 +···+in =2

i1 i2 i3 in ˙θ = 1 br + (bi1 ...in cos θ − ai1 ...in sin θ)(r cos θ) (r sin θ) z3 · · · zn +O(3) , r i1 +···+in =2 (k) ci1 ...in (r cos θ)i1 (r sin θ)i2 z3i3 · · · znin + O(3), k = 3, . . . , n, z˙k = εck zk + i1 +···+in =2

(1.13) where O(3) = O3 (r, z3 , . . . , zn ). As usual, Z+ denotes the set of all non-negative integers. Taking a00eij = has the sum of the entries equal to 2, it is easy to b00eij = 0 where eij ∈ Zn−2 + show that in a suitably small neighborhood of (r, z3 , . . . , zn ) = (0, 0, . . . , 0) we

1.2. Introduction: the classical theory

7

have θ˙ = 0. Then, choosing θ as the new independent variable system (1.13), in a neighborhood of (r, z3 , . . . , zn ) = (0, 0, . . . , 0) it becomes r dr = dθ

εar + br +

i

in (ai1 ...in cos θ + bi1 ...in sin θ)(r cos θ)i1 (r sin θ)i2 z33 · · · zn + O(3)

i1 +···+in =2

r dzk = dθ br +

εck zk +

,

i

in (bi1 ...in cos θ − ai1 ...in sin θ)(r cos θ)i1 (r sin θ)i2 z33 · · · zn + O(3)

i1 +···+in =2

i1 +···+in =2

(k) (r 1 ...in

ci

i

in cos θ)i1 (r sin θ)i2 z33 · · · zn + O(3) i

in (bi1 ...in cos θ − ai1 ...in sin θ)(r cos θ)i1 (r sin θ)i2 z33 · · · zn + O(3)

,

i1 +···+in =2

(1.14) for k = 3, . . . , n. We note that this system is 2π periodic in the variable θ. In order to write system (1.14) in the normal form of the averaging theory we rescale the variables (r, z3 , . . . , zn ) = (ρε, η3 ε, . . . , ηn ε).

(1.15)

Then the system (1.14) becomes dρ = εf1 (θ, ρ, η3 , . . . , ηn ) + ε2 g1 (θ, ρ, η3 , . . . , ηn , ε), dθ dηk = εfk (θ, ρ, η3 , . . . , ηn ) + ε2 gk (θ, ρ, η3 , . . . , ηn , ε), dθ

(1.16) k = 3, . . . , n,

where 1 f1 = b fk =

1 b

aρ +

(ai1 ...in cos θ + bi1 ...in sin θ)(ρ cos θ)

i1

i1 +···+in =2

cηk +

(ρ sin θ)i2 z3i3

· · · znin

,

(k)

i1 +···+in =2

ci1 ...in (ρ cos θ)i1 (ρ sin θ)i2 z3i3 · · · znin

.

We note that the system (1.16) is in the normal form (1.3) of the averaging theory, with x = (ρ, η3 , . . . , ηn ), t = θ, F (θ, ρ, η3 , . . . , ηn ) = (f1 (θ, ρ, η3 , . . . , ηn ), f3 (θ, ρ, η3 , . . . , ηn ), . . . , fn (θ, ρ, η3 , . . . , ηn )), and T = 2π. The averaged system of (1.16) is y˙ = εf 0 (y), y = (ρ, η3 , . . . , ηn ) ∈ Ω, (1.17) where Ω is a suitable neighborhood of the origin (ρ, η3 , . . . , ηn ) = (0, 0, . . . , 0), and f 0 (y) = (f10 (y), f30 (y), . . . , fn0 (y)), with fi0 (y) =

1 2π

2π

fi (θ, ρ, η3 , . . . , ηn )dθ, 0

i = 1, 3, . . . , n.

8

Chapter 1. The Averaging Theory for Computing Periodic Orbits

After some calculations we have that ⎞ ⎛ n 1 (a10ej + b01ej )ηj ⎠ , f10 = ρ ⎝2a + 2b j=3 ⎛ 1 (k) 2 ⎝2ck ηk + c(k) fk0 = + c 200n−2 020n−2 ρ + 2 2b

⎞

c00eij ηi ηj ⎠, (k)

k = 3, . . . , n,

3≤i≤j≤n

where ej ∈ Zn−2 is the unit vector with the j-th entry equal to 1, and eij ∈ Zn−2 + + has the sum of the i-th and j-th entries equal to 2 and the other equal to 0. Now we shall apply Theorem 1.2.1 for studying the limit cycles of system (1.16). Note that these limits, after the rescaling (1.15), will become inﬁnitesimal limit cycles for system (1.14), which will tend to the origin when ε 0; consequently, they will be bifurcated limit cycles of the Hopf bifurcation of system (1.14) at the origin. From Theorem 1.2.1 for studying the limit cycles of system (1.16) we only need to compute the non-degenerate singularities of system (1.17). Since the transformation from the cartesian coordinates (r, z3 , . . . , zn ) to the cylindrical ones (ρ, η3 , . . . , ηn ) is not a diﬀeomorphism at ρ = 0, we deal with the zeros having the coordinate ρ > 0 of the averaged function f 0 . So, we need to compute the roots of the algebraic equations 2a +

n

(a10ej + b01ej )ηj = 0, (k) (k) 2ck ηk + c200n−2 + c020n−2 ρ2 + 2 j=3

(k)

3≤i≤j≤n

c00eij ηi ηj = 0,

k = 3, . . . , n.

(1.18) Since the coeﬃcients of system (1.18) are independent and arbitrary, in order to simplify the notation we write it as a+

n

aj ηj = 0,

(k)

c0 ρ2 + ck ηk +

j=3

(k)

cij ηi ηj = 0,

k = 3, . . . , n, (1.19)

3≤i≤j≤n (k)

(k)

where aj , c0 , ck and cij are arbitrary constants. Denote by C the set of algebraic systems of form (1.19). We claim that there is a system belonging to C which has exactly 2n−3 simple roots. The claim can be veriﬁed by the example: a + a3 η3 = 0, (3) c 0 ρ2

+ c3 η3 +

ck ηk +

3≤i≤j≤k

(1.20) (3) cij ηi ηj

= 0,

(1.21)

3≤i≤j≤n (k)

cij ηi ηj = 0,

k = 4, . . . , n,

(1.22)

1.2. Introduction: the classical theory

9

with all the coeﬃcients being non-zero. Equations (1.22) can be treated as quadratic algebraic equations in ηk . Substituting the unique solution η30 of η3 in (1.20) into (1.22) with k = 4, this last equation has exactly two diﬀerent solutions, namely η41 and η42 for η4 , choosing conveniently c4 . Introducing the two solutions (η30 , η4i ), i = 1, 2, into (1.22) with k = 5 and choosing conveniently the values of the coeﬃcients of equation (1.22) with k = 5 and (η3 , η4 ) = (η30 , η4i ), we get two diﬀerent solutions η5i1 and η5i2 of η5 for each i. Moreover, playing with the coeﬃcients of the equations, the four solutions (η30 , η4i , η5ij ) for i, j = 1, 2, are distinct. By induction, we can prove that for suitable choice of the coeﬃcients, equations (1.20) and (1.22) have 2n−3 diﬀerent roots (η3 , . . . , ηn ). Since η3 = η30 is ﬁxed, (3) (3) for any given cij there exist values of c3 and c0 such that equation (1.21) has a n−3 solutions (η3 , . . . , ηn ) of (1.20) and (1.22). positive solution ρ for each of the 2 Since the 2n−3 solutions are diﬀerent, and the number of the solutions of (1.20)– (1.22) is the maximum that the equations can have (by the Bezout Theorem, see for instance [80]), it follows that every solution is simple, and consequently the determinant of the Jacobian of the system evaluated at it is not zero. This proves the claim. Using the same arguments which allowed us to prove the claim, we can also prove that we can choose the coeﬃcients of the previous system in order to have 0, 1, . . . , 2n−3 − 1 simple real solutions. Taking the averaged system (1.17) with f 0 having the convenient coeﬃcients as in (1.20)–(1.22), the averaged system (1.17) has exactly k ∈ {0, 1, . . . , 2n−3 } singularities with the components ρ > 0. Moreover, the determinants of the Jacobian matrix ∂f 0 /∂y at these singularities do not vanish because all the singularities are simple. In short, by Theorem 1.2.1 we get that there are systems of the form (1.11) which have k ∈ {0, 1, . . . , 2n−3 } limit cycles. This proves the theorem. An application to Hamiltonian systems The results of this subsubsection come from the paper Guirao–Llibre–Vera [39]. We consider the following class of Hamiltonians in the action-angle variables H(I1 , . . . , In , θ1 , . . . , θn ) = H0 (I1 ) + εH1 (I1 , . . . , In , θ1 , . . . , θn ),

(1.23)

where ε is a small parameter. For more details on the action-angle variables see, for instance, [1]. As usual, the Poisson bracket of the functions f (I1 , . . . , In , θ1 , . . . , θn ) and g(I1 , . . . , In , θ1 , . . . , θn ) is {f, g} =

n ∂f ∂g ∂f ∂g − . ∂θi ∂Ii ∂Ii ∂θi i=1

The next result provides suﬃcient conditions for computing periodic orbits of the Hamiltonian system associated to the Hamiltonian (1.23).

10

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Theorem 1.2.5. We deﬁne 1 H1 = 2π

2π H1 (I1 , . . . , In , θ1 , . . . , θn )dθ1 , 0

and we consider the diﬀerential system dIi {Ii , H1 } = ε −1 ∗ = εfi−1 (I2 , . . . , In , θ2 , . . . , θn ), dθ1 H0 (H0 (h ))

i = 2, ..., n, (1.24)

dθi {θi , H1 } = ε −1 ∗ = εfi+n−2 (I2 , . . . , In , θ2 , . . . , θn ), dθ1 H0 (H0 (h ))

i = 2, . . . , n,

restricted to the energy level H = h∗ with h∗ ∈ R. The value h∗ is such that the function H0−1 in a neighborhood of h∗ is a diﬀeomorphism. The system (1.24) is a Hamiltonian system with Hamiltonian ε H1 . If ε = 0 is suﬃciently small then for every equilibrium point p = (I20 , . . . , In0 , θ20 , . . . , θn0 ) of system (1.24) satisfying that ∂(f1 , . . . , f2n−2 ) = 0, det ∂(I2 , . . . , In , θ2 , . . . , θn ) (I2 ,...,In ,θ2 ,...,θn )=(I 0 ,...,I 0 ,θ0 ,...,θ0 ) 2

n

2

n

there exists a 2π-periodic solution γε (θ, . . . , In (θ1 , ε), θ2 (θ1 , ε), . . ., θn (θ1 , ε)) of the Hamiltonian system associated to the Hamiltonian (1.23), taking as independent variable the angle θ1 such that γε (0) → (H0−1 (h∗ ), I20 , . . . , In0 , θ20 , . . . , θn0 ) when ε → 0. The stability or instability of the periodic solution γε (θ1 ) is given by the stability or instability of the equilibrium point p of system (1.24). In fact, the equilibrium point p has the stability behavior of the Poincar´e map associated to the periodic solution γε (θ1 ). Now we clarify some of the notations used in the statement of Theorem 1.2.5. The function H0 is only a function of the variable I1 , i.e., H0 : J → R where J is an open subset of R (the domain of deﬁnition of H0 ), and consequently H0 (I1 ) ∈ R. Therefore, H0 means derivative with respect to the variable I1 . The diﬀerential system (2) is deﬁned on the energy level H(I1 , . . . , In , θ1 , . . . , θn ) = h∗ with h∗ ∈ R, and we assume that the value h∗ is such that the function H0−1 in a neighborhood of h∗ is a diﬀeomorphism. Therefore, the expression H0 (H0−1 (h∗ )) is well deﬁned. On the other hand, every periodic solution of a diﬀerential system has deﬁned in its neighborhood a return map F usually called the Poincar´e map. The periodic solution provides a ﬁxed point of the map F . The stability or instability of this ﬁxed point for the map F is what we call the stability behavior of the Poincar´e map associated to the periodic solution in the statement of Theorem 1.2.5. For more details on the Poincar´e map see, for instance, [76]. Theorem 1.2.5 will be proved later on.

1.2. Introduction: the classical theory

11

The next goal is to study the periodic orbits of the Hamiltonian system with the perturbed Keplerian Hamiltonian of the form H=

1 2 1 P1 + P22 + P32 − 2 + εP1 (Q21 + Q22 , Q3 ). 2 Q1 + Q22 + Q23

(1.25)

Note that the perturbation is symmetric with respect to the Q3 -axis. It is easy to check that the third component K = Q1 P2 − Q2 P1 of the angular momentum is a ﬁrst integral of the Hamiltonian system associated to the Hamiltonian (1.25). We use this second ﬁrst integral to simplify the analysis of the given axially symmetric Keplerian perturbed system. In the following we use the Delaunay variables for studying easily the periodic orbits of the Hamiltonian system associated to the Hamiltonian (1.25), see [23, 71] for more details on the Delaunay variables. Thus, in Delaunay variables, the Hamiltonian (1.25) has the form H=−

1 1 + εP(l, g, k, L, G, K) = − 2 + εP(l, g, L, G, K), 2L2 2L

(1.26)

where l is the mean anomaly, g is the argument of the perigee of the unperturbed elliptic orbit measured in the invariant plane, k is the longitude of the node, L is the square root of the semi-major axis of the unperturbed elliptic orbit, G is the modulus of the total angular momentum, and K is the third component of the angular momentum. Moreover, P is the perturbation obtained from the perturbation P1 using the transformation to Delaunay variables, namely Q1 = r (cos(f + g) cos k − c sin(f + g) sin k) , Q2 = r (cos(f + g) sin k + c sin(f + g) cos k) ,

(1.27)

Q3 = rs sin(f + g), with

K2 K , s2 = 1 − 2 . G G The true anomaly f and the eccentric anomaly E are auxiliary quantities deﬁned by the relations c=

G 1 − e2 = , r = a(1 − e cos E), l = E − e sin E. L √ a(cos E − e) a 1 − e2 sin E , cos f = , sin f = r r where e is the eccentricity of the unperturbed elliptic orbit. Note that the angular variable k is a cyclic variable for the Hamiltonian (1.26) and, consequently, K is a ﬁrst integral of the Hamiltonian system as we already knew.

12

Chapter 1. The Averaging Theory for Computing Periodic Orbits

The family of Hamiltonians (1.26) is a particular subclass of the Hamiltonians (1.23) with H1 = P. We denote by P the averaged map of P with respect to the mean anomaly l, i.e., 1 P = 2π

2π

1 P(l, g, L, G, K)dl = 2π

2π P(E − e sin E, g, L, G, K)(1 − e cos E)dE. 0

We remark that the map P only depends on the angle g and the three action variables L, G, K. We claim that H0 (H0−1 (h∗ )) = (−2h∗ )3/2 . Indeed, H0 (L) = −1/(2L2) = h∗ so, H0−1 (h∗ ) = (−2h∗ )1/2 . Since H0 (L) = 1/L3, the claim follows. From the deﬁnition of Poisson parenthesis, we also have that {G, P } = − {g, P } =

∂ P

∂G ∂ P

=− , ∂G ∂g ∂g

∂g ∂ P

∂ P

= , ∂g ∂G ∂G

∂k ∂ P

∂ P

= . ∂k ∂K ∂K Then, by Theorem 1.2.5 at the energy level H = h∗ with h∗ < 0 (because H0 (L) = −1/(2L2)) and with angular momentum K = k ∗ , the diﬀerential system (1.24) with respect to the mean anomaly l is {k, P } =

{G, P } ∂ P

dG = ε −1 ∗ = −ε(−2h∗ )3/2 = −εf1 (g, G, K), dl ∂g H0 (H0 (h )) {g, P } ∂ P

dg = ε −1 ∗ = ε(−2h∗ )3/2 = εf2 (g, G, K), dl ∂G H0 (H0 (h ))

(1.28)

{k, P } ∂ P

dk = ε −1 ∗ = ε(−2h∗ )3/2 = εf3 (g, G, K). dl ∂K H0 (H0 (h )) Note that we do not write the diﬀerential equation dK/dt = 0 because we are working in the invariant set H = h∗ and K = k ∗ . Now, we are ready to state a corollary of Theorem 1.2.5 providing suﬃcient conditions for the existence and the kind of stability of the periodic orbits in the perturbed Kepler problems with axial symmetry. Corollary 1.2.6. System (1.28) is the Hamiltonian system taking as independent variable the mean anomaly l of the Hamiltonian (1.25) written in Delaunay variables on the ﬁxed energy level H = h∗ < 0 and on the ﬁxed third component of the angular momentum K = k ∗ . If ε = 0 is suﬃciently small then, for every solution p = (g0 , G0 , k ∗ ) of the system fi (g, G, K) = 0 for i = 1, 2, 3 satisfying ∂(f1 , f2 , f3 ) det = 0, (1.29) ∂(g, G, K) (g,G,K)=(g0 ,G0 ,k∗ )

1.2. Introduction: the classical theory

13

and all k0 ∈ [0, 2π) there exists a 2π-periodic solution γε (l) = g(l, ε), k(l, ε), L(l, ε), G(l, ε), K(l, ε) = k ∗ √ such that γε (0) → (g0 , k0 , −2h∗ , G0 , k ∗ ) when ε → 0. The stability or instability of the periodic solution γε (l) is given by the stability or instability of the equilibrium point p of system (1.28). In fact, the equilibrium point p has the stability behavior of the Poincar´e map associated to the periodic solution γε (l). We remark that having a periodic solution for every k0 ∈ [0, 2π) with the same initial conditions for all of the other variables, means that we really have a two dimensional torus foliated by periodic solutions. There are many papers studying periodic orbits of diﬀerent perturbed Keplerian problems, see for instance [42, 47, 79] and the papers quoted therein. In what follows we shall study the spatial generalized van der Waals Hamiltonian system modeling the dynamical symmetries of the perturbed hydrogen atom. The generalized van der Waals Hamiltonian system was proposed in the paper [3] via the following Hamiltonian with β ∈ R H=

1 1 2 P1 + P22 + P32 − 2 + ε Q21 + Q22 + β 2 Q23 . 2 2 2 Q1 + Q2 + Q3

(1.30)

Note that this Hamiltonian is of the form (1.25). For more references, see the ones quoted in [38]. Theorem 1.2.7. On every energy level H = h∗ < 0, and for the third component of the angular momentum K = k ∗ , the spatial van der Waals Hamiltonian system associated to the Hamiltonian (1.30) for ε = 0 suﬃciently small has: (i) For K = k ∗ = 0, two 2π-periodic solutions γε± (l) = g(l, ε), k(l, ε)), L(l, ε), G(l, ε), K(l, ε) such that

3(β 2 + 1) 1 1 1 √ √ γε± (l)(0) → ± arccos , , , 0 , k 0 2 5(β 2 − 1) −2h∗ −2h∗ when ε → 0, for each k0 ∈ [0, 2π) if β ∈ (−∞, −2) ∪ (−1/2, 1/2) ∪ (2, ∞). These periodic orbits have a stable manifold of dimension 2 and an unstable one of dimension 1 if β ∈ (−1/2, 1/2), and have a stable manifold of dimension 1 and an unstable one of dimension 2 if β ∈ (−∞, −2) ∪ (2, ∞). Consequently, these periodic orbits are unstable. (ii) For K = k ∗ = 0, four 2π-periodic solutions γε±,± (l) = g(l, ε), k(l, ε)), L(l, ε), G(l, ε), K(l, ε) such that 5 5(1 − 4β 2 ) 1 1 π 1 ±,± ,± , γε (0) → ± , k0 , √ 2 4 −2h∗ (1 − β 2 ) −2h∗ 2 −2h∗ when ε → 0, for each k0 ∈ [0, 2π) if β ∈ (−1, −1/2) ∪ (1/2, 1).

14

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Theorem 1.2.7 is proved later on. The result in statement (i) was already obtained using cylindrical coordinates in [38]. The stability or instability of the four periodic orbits in statement (ii) can be determined analyzing the eigenvalues of the corresponding Jacobian matrices, but since the expression of these eigenvalues are huge and depend on the two parameters h∗ and β, this study is a long task we are not going to do here. We remark that, when (β 2 − 1)(β 2 − 4)(β 2 − 1/4) = 0, i.e., for the values that the averaging theory for ﬁnding periodic orbits do not provide any information, it is known that the van der Waals Hamiltonian system is integrable, see [33]. Therefore the averaging method, when it cannot be applied for ﬁnding periodic orbits, provides a suspicion that for such values of the parameter the system could be integrable. The Hamiltonian system associated to the Hamiltonian (1.23) can be written as ∂H1 dIi = ε{Ii , H1 } = −ε , i = 1, . . . , n, dt ∂θi ∂H1 dθi = ε{θi , H1 } = ε , i = 2, . . . , n, (1.31) dt ∂Ii ∂H1 dθ1 = H0 (I1 ) + ε{θ1 , H1 } = H0 (I1 ) + ε . dt ∂I1 Lemma 1.2.8. Taking as new independent variable the variable θ1 , we have in the ﬁxed energy level H = h∗ < 0 that the diﬀerential system (1.31) becomes dIi {Ii , H1 } = ε −1 ∗ + O(ε2 ), dθ1 H0 (H0 (h ))

i = 2, . . . , n,

dθi {θi , H1 } = ε −1 ∗ + O(ε2 ), dθ1 H0 (H0 (h ))

i = 2, . . . , n,

(1.32)

with I1 = H0−1 (h∗ ) + O(ε) if H0 (H0−1 (h∗ )) = 0. Proof. Taking as new independent variable θ1 , equations (1.31) become {Ii , H1 } ε{Ii , H1 } dIi = =ε + O(ε2 ), dθ1 H0 (I1 ) + ε{θ1 , H1 } H0 (I1 )

i = 1, . . . , n,

{θi , H1 } dθi ε{θi , H1 } = =ε + O(ε2 ), dθ1 H0 (I1 ) + ε{θ1 , H1 } H0 (I1 )

i = 2, . . . , n.

Fixing the energy level of H = h∗ < 0, we obtain h∗ = H0 (I1 ) + εH1 (I1 , . . . , In , θ1 , . . . , θn ). Using the Implicit Function Theorem and the fact that H0 (H0−1 (h∗ )) = 0, for ε suﬃciently small, we get I1 = H0−1 (h∗ )+ O(ε), and the equations are reduced to (1.32).

1.2. Introduction: the classical theory

15

Proof of Theorem 1.2.5. The averaged system in the angle θ1 obtained from (1.32) is 2π 1 dIi ∂H1 ε =− dθ1 , i = 2, . . . , n, −1 dθ1 2π H0 (H0 (h)) ∂θi 0

2π dθi {θi , H1 } ∂H1 1 ε = dθ1 , dθ1 2π H0 (H0−1 (h)) ∂Ii

(1.33) i = 2, . . . , n.

Since 1 ∂ H1

= ∂θi 2π

2π ∂H1 dθ1 , ∂θi

i = 2, . . . , n,

∂ H1

1 = ∂Ii 2π

2π ∂H1 dθ1 , ∂Ii

i = 2, . . . , n,

the diﬀerential system (1.33) becomes dIi ∂ H1

ε {Ii , H1 } , = − −1 = ε −1 dθ1 H0 (H0 (h)) ∂θi H0 (H0 (h)) dθi ∂ H1

ε {θi , H1 } , = −1 = ε −1 dθ1 ∂I H0 (H0 (h)) H0 (H0 (h)) i

i = 2, . . . , n, i = 2, . . . , n,

which coincides with (1.24). Once we have obtained the averaged system (1.24), it is immediate to check that it satisﬁes the assumptions of Theorem 1.2.1, then applying the conclusions of this theorem the rest of the statement of Theorem 1.2.5 follows immediately. Proof of Theorem 1.2.7. For the generalized van der Waals Hamiltonian system, the function P(E, g, h, G, K) is equal to 2 2 β G + G2 + K 2 − K 2 β 2 (e cos E − 1)2 L4 2G2 4 2 2 2 L (G − K )(β − 1)(e − cos E)2 cos2 g − 2G2 4 2 2 2 L (G − K )(β − 1)(e − cos E)2 sin2 g + 2G2 3 2 2 2 2L (G − K )(β − 1)(e − cos E) cos g sin E sin g − G 1 2 2 2 2 + L (G − K )(β − 1) cos2 g sin2 E 2 1 − L2 (G2 − K 2 )(β 2 − 1) sin2 E sin2 g. 2

16

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Its averaged function with respect to the mean anomaly is 1 P = 2π

2π P(E, g, h, G, K)(1 − e cos E)dE =

B , 4G2

where B = L2 5(G2 − K 2 )(G2 − L2 )(β 2 − 1) cos(2g) − (3G2 − 5L2 )(G2 + K 2 + (G2 − K 2 )β 2 ) . Equations (1.28) are the averaged equations of the Hamiltonian system with Hamiltonian (1.30) dG 5(1 + 2h∗ G2 )(G2 − K 2 )(β 2 − 1) sin(2g) √ =ε = −εf1 (g, G, K), dl 2G2 −2h∗ dg C = −ε 3 √ = εf2 (g, G, K), dl 2G −2h∗ K(β 2 − 1)(−5 − 6h∗ G2 + 5(1 + 2h∗ G2 ) cos(2g)) dk √ =ε = εf3 (g, G, K), dl 2G2 −2h∗ where C√= 5K 2 (β 2 − 1) + 6h∗ G4 (β 2 + 1) − 5(2h∗ G4 + K 2 )(β 2 − 1) cos(2g); here, L = 1/ −2h∗ + O(ε). The equilibrium solutions (g0 , G0 , k ∗ ) of this averaged system satisfying (1.29) give rise to periodic orbits of the Hamiltonian system with Hamiltonian (1.30) for each H = h∗ < 0 and K = k ∗ , see Theorem 1.2.1. These equilibria (g0 , G0 , k ∗ ) are

3(β 2 + 1) 1 5 5(1 − 4β 2 ) π 1 1 1 . ,± ,√ ,0 , ± , ± arccos 2 5(β 2 − 1) 2 2 −2h∗ 4 −2h∗ (1 − β 2 ) −2h∗ The ﬁrst two equilibria exist if 3(β 2 + 1)/(5(β 2 − 1)) ∈ [−1, 1], i.e., if β ∈ (−∞, −2] ∪ [−1/2, 1/2] ∪ [2, ∞). √ The Jacobian (1.29) of the ﬁrst equilibrium is equal to J = 16 −2h∗ (β 2 − 1) (β 2 − 4)(β 2 − 1/4). So, when β ∈ (−∞, −2) ∪ (−1/2, 1/2) ∪ (2, ∞), each of these equilibria provides one periodic orbit of the Hamiltonian system with Hamiltonian (1.30) for each H = h∗ < 0 and K = k ∗ = 0. Since k ∗ = 0, these periodic orbits bifurcate from an elliptic orbit (g0 = 0) of the Kepler problem living in the plane of motion of the two bodies of the Kepler problem. Moreover, since the eigenvalues of the Jacobian matrix at these equilibra are ±2 (β 2 − 4)(4β 2 − 1) √ 2 ∗ and −2h (β − 1), these periodic orbits have a stable manifold of dimension 2 and an unstable one of dimension 1 if β ∈ (−1/2, 1/2), and have a stable manifold of dimension 1 and an unstable one of dimension 2 if β ∈ (−∞, −2) ∪ (2, ∞). This proves statement (i) of the theorem. The last four equilibria exist if β ∈ (−1, −1/2] ∪ [1/2, 1) and have Jacobian √ equal to J = −15 −2h∗ (β 2 − 1)(4β 2 − 1). So, for each value of k ∈ [0, 2π) these four equilibria when β ∈ (−1, −1/2) ∪ (1/2, 1) provide four periodic orbits

1.2. Introduction: the classical theory

17

of the Hamiltonian system with Hamiltonian (1.30) for each H = h∗ < 0 and 5(1 − 4β 2 ) 1 K = k∗ = ± = 0. Since k ∗ = 0 these periodic orbits bifurcate 4 −2h∗ (1 − β 2 ) from elliptic orbits (g0 = 0) of the Kepler problem which are not in the plane of motion deﬁned by the two bodies. This proves statement (ii) of the theorem.

1.2.3 Other ﬁrst order averaging methods for periodic orbits We consider the problem of bifurcation of T -periodic solutions from the diﬀerential system (1.34) x˙ = F0 (t, x) + εF1 (t, x) + ε2 R(t, x, ε), with ε = 0 to ε = 0 suﬃciently small. Here, the functions F0 , F1 : R × D → Rn and R : R × D × (−ε0 , ε0 ) → Rn are C 2 functions, T -periodic in the ﬁrst variable, and D is an open subset of Rn . One of the main assumptions is that the unperturbed system (1.35) x = F0 (t, x), has a submanifold of periodic solutions. Let x(t, z) be the solution of the unperturbed system (1.35) satisfying that x(0, z) = z. We write the linearization of the unperturbed system along the periodic solution x(t, z) as (1.36) y = Dx F0 (t, x(t, z))y. In what follows we denote by Mz (t) some fundamental matrix of the linear differential system (1.36), and by ξ : Rk × Rn−k → Rk the projection of Rn onto its ﬁrst k coordinates, i.e., ξ(x1 , . . . , xn ) = (x1 , . . . , xk ). The next result goes back to Malkin [29] and Roseau [76]. Here, we shall present the shorter proof given by A. Buic˘ a–Fran¸coise–Llibre [12]. Theorem 1.2.9. Let V ⊂ Rk be open and bounded, and let β0 : Cl(V ) → Rn−k be a C 2 function. We assume that (i) Z = {zα = (α, β0 (α)) : α ∈ Cl(V )} ⊂ Ω and that for each zα ∈ Z the solution x(t, zα ) of (1.35) is T -periodic; (ii) for each zα ∈ Z there is a fundamental matrix Mzα (t) of (1.36) such that (0) − Mz−1 (T ) has in the right up corner the k × (n − k) zero the matrix Mz−1 α α matrix, and in the right lower corner a (n − k) × (n − k) matrix Δα with det(Δα ) = 0. We consider the function F : Cl(V ) → Rk deﬁned as T

F (α) = ξ 0

Mz−1 (t)F1 (t, x(t, zα ))dt α

.

(1.37)

If there exists a ∈ V with F (a) = 0 and det ((dF /dα) (a)) = 0, then there is a T -periodic solution x(t, ε) of system (1.34) such that x(0, ε) → za as ε → 0.

18

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Theorem 1.2.9 is proved in Subsection 1.2.7. In the next subsection we provide some applications of this theorem. We assume that there exists an open set V with Cl(V ) ⊂ Ω such that for each z ∈ Cl(V ), x(t, z, 0) is T -periodic, where x(t, z, 0) denotes the solution of the unperturbed system (1.35) with x(0, z, 0) = z. The set Cl(V ) is isochronous for the system (1.34), i.e., it is a set formed only by periodic orbits, all of them having the same period. Then, an answer to the problem of the bifurcation of T -periodic solutions from the periodic solutions x(t, z, 0) contained in Cl(V ) is given in the following result. Corollary 1.2.10 (Perturbations of an isochronous set). We assume that there exists an open and bounded set V with Cl(V ) ⊂ Ω and such that, for each z ∈ Cl(V ), the solution x(t, z) is T -periodic; then we consider the function F : Cl(V ) → Rn , F (z) = 0

T

Mz−1 (t, z)F1 (t, x(t, z))dt.

(1.38)

If there exists a ∈ V with F (a) = 0 and det ((dF /dz) (a)) = 0, then there exists a T -periodic solution x(t, ε) of system (1.34) such that x(0, ε) → a as ε → 0. Proof. It follows immediately from Theorem 1.2.9 taking k = n.

1.2.4 Three applications In this subsection we shall develop three applications of Theorem 1.2.9 and of its Corollary 1.2.10. The Hopf bifurcation of the Michelson system The Michelson system x˙ = y,

y˙ = z,

z˙ = c2 − y −

x2 , 2

(1.39)

with (x, y, z) ∈ R3 and the parameter c ≥ 0, was introduced by Michelson [72] in the study of the travelling wave solutions of the Kuramoto–Sivashinsky equation. It is well known that system (1.39) is reversible with respect to the involution R(x, y, z) = (−x, y, −z) and is volume-preserving under the ﬂow of the system. It √ = (− 2c, 0, 0) is easy to check that system (1.39) has two ﬁnite singularities S 1 √ and S2 = ( 2c, 0, 0) for c > 0, which are both saddle-foci. The former has a two dimensional stable manifold and the latter has a two dimensional unstable manifold. For c > 0 small numerical experiments (see for instance Kent–Elgin [49]) and asymptotic expansions in sinus series (see Michelson [72] in 1986 and Webster– Elgin [86] in 2003) revealed the existence of a zero-Hopf bifurcation at the origin for c = 0. But their results do not provide an analytic proof on the existence of

1.2. Introduction: the classical theory

19

such zero-Hopf bifurcation. By a zero-Hopf bifurcation we mean that when c = 0 the Michelson system has the origin as a singularity having eigenvalues 0, ±i, and when c > 0 suﬃciently small the Michelson system has a periodic orbit which tends to the origin when c tends to zero. The analytic proof of this zero-Hopf bifurcation has been provided by Llibre–Zang [59]. Now we state this result and reproduce its proof. Theorem 1.2.11. For c ≥ 0 suﬃciently small the Michelson system (1.39) has a zero-Hopf bifurcation at the origin for c = 0. Moreover, the bifurcated periodic orbit satisﬁes x(t) = −2c cos t+o(c), y(t) = 2c sin t+o(c) and z(t) = 2c cot t+o(c), for c > 0 suﬃciently small. Proof. For any ε = 0 we apply the change of variables x = εx, y = εy, z = εz and c = εd, and Michelson system (1.39) becomes x˙ = y,

y˙ = z,

1 z˙ = −y + εd2 − ε x2 , 2

(1.40)

where we still use x, y, z instead of x, y, z. Now doing the change of variables x = x, y = r sin θ and z = r cos θ, system (1.40) goes over to x˙ = r sin θ,

r˙ =

ε (2d2 − x2 ) cos θ, 2

ε θ˙ = 1 − (2d2 − x2 ) sin θ. 2r

(1.41)

This system can be written as dx ε = r sin θ + (2d2 − x2 ) sin2 θ + ε2 f1 (θ, r, ε), dθ 2 ε dr = (2d2 − x2 ) cos θ + ε2 f2 (θ, r, ε), dθ 2

(1.42)

where f1 and f2 are analytic functions in their variables. For arbitrary (x0 , r0 ) = (0, 0), the system (1.42)ε=0 has the 2π-periodic solution x(θ) = r0 + x0 − r0 cos θ, r(θ) = r0 , (1.43) such that x(0) = x0 and r(0) = r0 . It is easy to see that the ﬁrst variational equation of (1.42)ε=0 along the solution (1.43) is ⎛ ⎞ dy1

0 sin θ y1 ⎜ dθ ⎟ . ⎝ dy2 ⎠ = 0 0 y2 dθ It has the fundamental solution matrix

1 1 − cos θ M= , 0 1

(1.44)

20

Chapter 1. The Averaging Theory for Computing Periodic Orbits

which is independent from the initial condition (x0 , r0 ). Applying Corollary 1.2.10 to the diﬀerential system (1.42) we have that 1 F (x0 , r0 ) = 2

2π M

−1

(2d2 − x2 ) sin2 θ (2d2 − x2 ) cos θ

dθ.

(1.43)

Then, F (x0 , r0 ) = (g1 (x0 , r0 ), g2 (x0 , r0 )) with g1 (x0 , r0 ) =

1 2 4d − 5r02 − 6r0 x0 − 2x20 , 4

g2 (x0 , r0 ) =

1 r0 (x0 + r0 ). 2

We can check that F = 0 has a unique non-trivial solution x0 = −2d and r0 = 2d, and that det DF (x0 , r0 )|x0 =−2d, r0 =2d = d2 . Hence by Corollary 1.2.10 it follows that, for any given d > 0 and for |ε| > 0 suﬃciently small, the system (1.42) has a periodic orbit (x(θ, ε), r(θ, ε)) of period 2π, such that (x(0, ε), r(0, ε)) → (−2d, 2d) as ε → 0. We note that the eigenvalues of DF (x0 , r0 )|x0 =−2d, r0 =2d are ±di. This shows that the periodic orbit is linearly stable. Going back to system (1.39) we get that, for c > 0 suﬃciently small, the Michelson system has a periodic orbit of period close to 2π given by x(t) = −2c cos t + o(c), y(t) = 2c sin t + o(c) and z(t) = 2c cos t + o(c). We think that this periodic orbit is symmetric with respect to the involution R, but we do not have a proof of it. A third-order diﬀerential equation Using Theorem 1.2.9 in the next result we present a third-order diﬀerential equation having as many limit cycles as we want. Proposition 1.2.12. Let us consider the third-order diﬀerential equation ... x − x¨ + x˙ − x = ε cos(x + t).

(1.45)

Then for all positive integer m there is εm > 0 such that if ε ∈ [−εm , εm ] \ {0} the diﬀerential equation (1.45) has at least m limit cycles. Proof. If y = x˙ and z = x ¨, then (1.45) can be written as x˙ = y, y˙ = z, z˙ = x − y + z + ε cos(x + t) = x − y + z + εF (t, x, y, z).

(1.46)

The origin (0, 0, 0) is the unique singular point of (1.46) when ε = 0. The eigenvalues of the linearized system at this singular point are ±i and 1. By the linear invertible transformation (X, Y, Z)T = C(x, y, z)T , where ⎛ ⎞ 1 −1 0 C = ⎝ 0 −1 1 ⎠ , 1 0 1

1.2. Introduction: the classical theory

21

we transform the diﬀerential system (1.46) into another such that its linear part is the real Jordan normal form of the linear part of system (1.46) with ε = 0, i.e., X˙ = −Y, Y˙ = X + εF˜ (X, Y, Z, t), Z˙ = Z + εF˜ (X, Y, Z, t),

(1.47)

where

F˜ (X, Y, Z, t) = F

X − Y + Z −X − Y + Z −X + Y + Z , , ,t . 2 2 2

Using the notation introduced in (1.34) we have that x = (X, Y, Z), F0 (x, t) = (−Y, X, Z), F1 (x, t) = (0, F˜ , F˜ ) and F2 (x, t) = 0. Let x(t; X0 , Y0 , Z0 , ε) be the solution to system (1.47) with x(0; X0 , Y0 , Z0 , ε) = (X0 , Y0 , Z0 ). Clearly the unperturbed system (1.47) with ε = 0 has a linear center at the origin in the (X, Y )plane, which is an invariant plane under the ﬂow of the unperturbed system, and the periodic solution x(t; X0 , Y0 , 0, 0) = (X(t), Y (t), Z(t)) is X(t) = X0 cos t − Y0 sin t,

Y (t) = Y0 cos t + X0 sin t,

Z(t) = 0.

(1.48)

Note that all these periodic orbits have period 2π. For our system, V and α from Theorem 1.2.9 are V = {(X, Y, 0) : 0 < X 2 + Y 2 < ρ}, for some arbitrary ρ > 0 and α = (X0 , Y0 ) ∈ V . The fundamental matrix solution M (t) of the variational equation of the unperturbed system (1.47)ε=0 with respect to the periodic orbits (1.48) satisfying that M (0) is the identity matrix is ⎛

cos t M (t) = ⎝ sin t 0

⎞ − sin t 0 cos t 0 ⎠. 0 et

We remark that it is independent from the initial condition (X0 , Y0 , 0). Moreover an easy computation shows that ⎛

0 M −1 (0) − M −1 (2π) = ⎝ 0 0

⎞ 0 0 ⎠. 0 0 0 1 − e−2π

In short we have shown that all the assumptions of Theorem 1.2.9 hold. Hence we shall study the zeros α = (X0 , Y0 ) ∈ V of the two components of the function F (α) given in (1.37). More precisely we have F (α) = (F1 (α), F2 (α)) where

22

Chapter 1. The Averaging Theory for Computing Periodic Orbits

2π

sin tF˜ (x(t; X0 , Y0 , 0, 0), t)dt

2π X(t) − Y (t) X(t) + Y (t) −X(t) + Y (t) = ,− , , t dt, sin tF 2 2 2 0 2π F2 (α) = cos tF˜ (x(t; X0 , Y0 , 0, 0), t)dt 0

2π X(t) − Y (t) X(t) + Y (t) −X(t) + Y (t) ,− , , t dt, cos tF = 2 2 2 0

F1 (α) =

where X(t), Y (t) are given by (1.48). First, we consider the third-order diﬀerential equation (1.45). For this equation we have that

2π (X0 − Y0 ) cos t − (X0 + Y0 ) sin t) dt, f1 (X0 , Y0 ) = sin t cos t + 2 0

2π (X0 − Y0 ) cos t − (X0 + Y0 ) sin t) cos t cos t + dt. f2 (X0 , Y0 ) = 2 0 To simplify the computation of these two integrals we do the change of variables (X0 , Y0 ) → (r, s) given by X0 − Y0 = 2r cos s,

X0 + Y0 = −2r sin s,

(1.49)

where r > 0 and s ∈ [0, 2π). From now on and until the end of the paper, we write f1 (r, s) instead of f1 (X0 , Y0 ) = f1 r(cos s − sin s), −r(cos s + sin s) . Similarly for f2 (r, s). We compute the two previous integrals and we get f1 (r, s) = −πJ2 (r) sin 2s,

1 J1 (r) − J2 (r) cos2 s , f2 (r, s) = 2π r

(1.50)

where J1 and J2 are the ﬁrst and second Bessel functions of the ﬁrst kind. For more details on Bessel functions, see [2]. These computations become easier with the help of an algebraic manipulator such as Mathematica or Maple. Using the asymptotic expressions of the Bessel functions of ﬁrst kind it follows that Bessel functions J1 (r) and J2 (r) have diﬀerent zeros. Hence, fi (r, s) = 0 for i = 1, 2 imply that s ∈ {0, π/2, π, 3π/2}. Therefore, we have to study the zeros of

1 J1 (r) − J2 (r) , f2 (r, 0) = f2 (r, π) = 2π (1.51) r

1.2. Introduction: the classical theory

23

2π J1 (r). (1.52) r We claim that function (1.51) has also inﬁnite zeros for r ∈ (0, ∞). Note that if ρ is suﬃciently large, and we choose r < ρ also suﬃciently large, then nπ π 2 Jn (r) ≈ cos r − − , for n = 1, 2, πr 2 4 f2 (r, π/2) = f2 (r, 3π/2) =

are asymptotic estimations, see [2]. Considering (1.51) for r suﬃciently large we obtain

2 2π 3π π f2 (r, 0) ≈ cos r − + r cos r − r r 4 4 2 π ((r − 1) cos r + (r + 1) sin r). = r r The above function has inﬁnite zeros because the equation tan r =

1−r r+1

has inﬁnitely many solutions. For every zero r0 > 0 of the function (1.51) we have two zeroes of system (1.50), namely (r, s) = (r0 , 0) and (r, s) = (r0 , π). We have from (1.50) that ∂(f1 , f2 ) 4π 2 (J0 (r0 )r0 − 2J1 (r0 ))(J0 (r0 )r0 + (r02 − 2)J1 (r0 )) = ∂(r, s) r03 (r,s)=(r0 ,0) =

4π 2 J2 (r0 )(J1 (r0 )r0 − J2 (r0 )), r0

(1.53)

where we have used several relations between the Bessel functions of the ﬁrst kind, see [2]. Clearly, it is impossible that (1.51) and (1.53) are equal to zero at the same time. Therefore, by Theorem 1.2.1, there is a periodic orbit of the system (1.45) for each (r0 , 0), that is, for each value of (X0 , Y0 ) = (r0 , −r0 ). In an analogous way, there is a periodic orbit of the system (1.45) for each (r0 , π), that is, for each value of (X0 , Y0 ) = (−r0 , r0 ). In fact, the periodic orbit with these initial conditions and the previous one with initial conditions (X0 , Y0 ) = (r0 , −r0 ) are the same. Similarly, since J1 (r) has inﬁnitely many zeroes (see [2]), the function (1.52) has inﬁnitely many positive zeroes r1 . Every one of these zeroes provides two solutions to the system (1.50), namely (r, s) = (r1 , π/2) and (r, s) = (r1 , 3π/2). Moreover we have from (1.50) that ∂(f1 , f2 ) 4π 2 2 = J (r1 ) = 0. (1.54) ∂(r, s) r1 2 (r,s)=(r1 ,π/2)

24

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Therefore, by Theorem 1.2.1, there is a periodic orbit of the system (1.45) for each (r1 , π/2), that is, for each value of (X0 , Y0 ) = (−r1 , −r1 ). In an analogous way there is also a periodic orbit of the system (1.45) for each (r1 , 3π/2), that is, for each value of (X0 , Y0 ) = (r1 , r1 ). In fact, the periodic orbit with these initial conditions and the previous one with initial conditions (X0 , Y0 ) = (−r1 , −r1 ) are the same. Taking the radius ρ of the disc V = {(X0 , Y0 , 0) : 0 < X 2 + Y 2 < ρ} in the proof of Theorem 1.2.9 conveniently large, we include in it as many zeros of the system f1 (X0 , Y0 ) = f2 (X0 , Y0 ) = 0 as we want, so from Theorem 1.2.9, Proposition 1.2.12 follows. The Vallis system (El Ni˜ no phenomenon) The results of this subsubsection come from the paper Euz´ebio–Llibre [32]. The Vallis system, introduced by Vallis [84] in 1988, is a periodic nonautonomous three dimensional system modeling the atmosphere dynamics in the tropics over the Paciﬁc Ocean, related to the yearly oscillations of precipitation, temperature and wind force. Denoting by x the wind force, by y the diﬀerence of near-surface water temperatures of the east and west parts of the Paciﬁc Ocean, and by z the average near-surface water temperature, the Vallis system is dx = −ax + by + au(t), dt dy = −y + xz, dt dz = −z − xy + 1, dt

(1.55)

where u(t) is some C 1 T -periodic function describing the wind force under seasonal motions of air masses, and the parameters a and b are positive. Although this model neglects some eﬀects like Earth’s rotation, pressure ﬁeld and wave phenomena, it provides a correct description of the observed processes and recovers many of the observed properties of El Ni˜ no. The properties of El Ni˜ no phenomena are studied analytically in [82, 84]. More precisely, in [84] it is shown that, taking u ≡ 0, it is possible to observe the presence of chaos by considering a = 3 and b = 102. Later on, in [82], it is proved that there exists a chaotic attractor for the system (1.55) after a Hopf bifurcation. This chaotic motion can be easily understood if we observe the strong similarity between the system (1.55) and the Lorenz system, which becomes more clear under the replacement of z by z + 1 in (1.55). Now we shall provide suﬃcient conditions in order that system (1.55) has periodic orbits and, additionally, we shall characterize the stability of these periodic orbits. As far as we know, the study of the periodic orbits in the non-autonomous Vallis system has not been considered in the literature, with the exception of the Hopf bifurcation studied in [82].

1.2. Introduction: the classical theory We deﬁne

I=

25

T

u(s)ds. 0

Now we state our main result. Theorem 1.2.13. For I = 0 and a = b the Vallis system (1.55) has a T -periodic solution (x(t), y(t), z(t)) such that

aI aI (x(t), y(t), z(t)) ≈ , ,1 . T (a − b) T (a − b) Moreover this periodic orbit is stable if a > b, and unstable if a < b. We do not know the reliability of the Vallis model approximating the Ni˜ no phenomenon but it seems that, for the moment, this is one of the best existing models. Accepting this reliability we can say the following. The stable periodic solution provided by Theorem 1.2.13 says that the Ni˜ no phenomenon exhibits a periodic behavior if the T -periodic function u(t) and the parameters a and b of the system satisfy I = 0 and a > b. Moreover, Theorem 1.2.13 states that this periodic solution lives near the point

aI aI (x, y, z) = , ,1 . T (a − b) T (a − b) Since the periodic solutions found in the following Theorems 1.2.15, 1.2.16 and 1.2.17 are also stable, we can provide a similar physical interpretation for them as we have done for the periodic solution from Theorem 1.2.13. Theorem 1.2.14. For I = 0 the Vallis system (1.55) has a T -periodic solution (x(t), y(t), z(t)) such that

aI aI (x(t), y(t), z(t)) ≈ − , − , 1 . Tb Tb Moreover this periodic orbit is always unstable. Theorem 1.2.15. For I = 0 the Vallis system (1.55) has a T -periodic solution (x(t), y(t), z(t)) such that

I I (x(t), y(t), z(t)) ≈ , ,1 . T T Moreover this periodic orbit is always stable. Theorem 1.2.16. For I = 0 the Vallis system (1.55) has a T -periodic solution (x(t), y(t), z(t)) such that

I (x(t), y(t), z(t)) ≈ , 0, 1 . T Moreover this periodic orbit is always stable.

26

Chapter 1. The Averaging Theory for Computing Periodic Orbits In what follows we consider the function

t

u(s)ds,

J(t) = 0

and note that J(T ) = I. So, we have the following result. Theorem 1.2.17. Consider I = 0 and J(t) = 0 if 0 < t < T . Then, the Vallis system (1.55) has a T -periodic solution (x(t), y(t), z(t)) such that (x(t), y(t), z(t)) ≈

a − T

T

J(s)ds, 0, 1 . 0

Moreover this periodic orbit is always stable. The tool for proving our results will be the averaging theory. This theory applies to periodic non-autonomous diﬀerential systems depending on a small parameter ε. Since the Vallis system already is a T -periodic non-autonomous diﬀerential system, in order to apply to it the averaging theory described in Section 1.4 we need to introduce in such system a small parameter. This is reached doing convenient rescalings in the variables (x, y, z), in the parameters (a, b), and in the function u(t). Playing with diﬀerent rescalings we shall obtain diﬀerent results on the periodic solutions of the Vallis system. More precisely, in order to study the periodic solutions of the diﬀerential system (1.55), we start doing a rescaling of the variables (x, y, z), of the function u(t), and of the parameters a and b, as follows: x = εm1 X,

y = εm2 Y,

z = εm3 Z,

u(t) = εn1 U (t),

a = εn2 A,

b = εn3 B,

(1.56)

where ε is always positive and suﬃciently small, and where mi and nj are nonnegative integers, for all i, j = 1, 2, 3. Then, in the new variables (X, Y, Z), the system (1.55) is written dX = −εn2 AX + ε−m1 +m2 +n3 BY + ε−m1 +n1 +n2 AU (t), dt dY = −Y + εm1 −m2 +m3 XZ, dt dZ = −Z − εm1 +m2 −m3 XY + ε−m3 . dt

(1.57)

Consequently, in order to have non-negative powers of ε we must impose the conditions m3 = 0

and 0 ≤ m2 ≤ m1 ≤ L,

(1.58)

1.2. Introduction: the classical theory

27

where L = min{m2 + n3 , n1 + n2 }. So, system (1.57) becomes dX = −εn2 AX + ε−m1 +m2 +n3 BY + ε−m1 +n1 +n2 AU (t), dt dY = −Y + εm1 −m2 XZ, dt dZ = 1 − Z − εm1 +m2 XY. dt

(1.59)

Our aim is to ﬁnd periodic solutions of the system (1.59) for some special values of mi , nj , i, j = 1, 2, 3, and after we go back through the rescaling (1.56) to guarantee the existence of periodic solutions in system (1.55). In what follows we consider the case where n2 and n3 are positive and m2 = m1 < n1 + n2 . These conditions lead to the proofs of Theorems 1.2.13, 1.2.14 and 1.2.15. For this reason we present these proofs together in order to avoid repetitive arguments. Moreover, in what follows we consider T

K=

U (s)ds. 0

Proofs of Theorems 1.2.13, 1.2.14 and 1.2.15. We start considering system (1.59) with n2 and n3 positive and m2 = m1 < n1 + n2 . So we have dX = −εn2 AX + εn3 BY + ε−m1 +n1 +n2 AU (t), dt dY = −Y + XZ, dt dZ = 1 − Z − ε2m1 XY. dt

(1.60)

Now we apply the averaging method to the diﬀerential system (1.60). Using the notation of Subsection 1.2.5, we have x= (X, Y, Z)T and ⎛ ⎞ 0 F0 (t, x) = ⎝ −Y + XZ ⎠ . (1.61) 1−Z We start considering the system x˙ = F0 (t, x).

(1.62)

Its solution x(t, z, 0) = (X(t), Y (t), Z(t)) such that x(0, z, 0) = z = (X0 , Y0 , Z0 ) is X(t) = X0 , Y (t) = (1 − e−t (1 + t))X0 + e−t Y0 + e−t tX0 Z0 , Z(t) = 1 − e−t + e−t Z0 .

28

Chapter 1. The Averaging Theory for Computing Periodic Orbits

In order that x(t, z, 0) is a periodic solution we must choose Y0 = X0 and Z0 = 1. This implies that, through every point of the straight line X = Y , Z = 1, there passes a periodic orbit lying in the phase space (X, Y, Z, t) ∈ R3 × S1 . Here and in what follows, S1 is the interval [0, T ] identifying T with 0. Observe that, using the notation of Subsection 1.2.5, we have n = 3, k = 1, α = X0 and β(X0 ) = (X0 , 1) and, consequently, M is a one dimensional manifold given by M = {(X0 , X0 , 1) ∈ R3 : X0 ∈ R}. The fundamental matrix Mz (t) of (1.62), satisfying that Mz (0) is the identity of R3 , is ⎞ ⎛ 1 0 0 ⎝ 1 − cosh t + sinh t e−t e−t tX0 ⎠ , 0 0 e−t and its inverse matrix Mz−1 (t) is ⎛

1 ⎝ 1 − et 0

0 et 0

⎞ 0 −et tX0 ⎠ . et

Since the matrix Mz−1 (0) − Mz−1 (T ) has an 1 × 2 zero matrix in the upper right corner, and a 2 × 2 lower right corner matrix

1 − e T e T T X0 Δ= , 0 1 − eT with det(Δ) = (1 − eT )2 = 0 because T = 0, we can apply the averaging theory described in Subsection 1.2.5. Let F be the vector ﬁeld of system (1.60) minus F0 given in (1.61). Then the components of the function Mz−1 (t)F (t, x(t, z, 0)) are g1 (X0 , t) = −εn2 AX0 + εn3 BX0 + ε−m1 +n1 +n2 AU (t), g2 (X0 , t) = ε2m1 et tX03 + (1 − et )g1 (X0 , t), g3 (X0 , t) = −ε2m1 et X02 . In order to apply averaging theory of ﬁrst order we need to consider only terms up to order ε. Analysing the expressions of g1 , g2 and g3 we note that these terms depend on the values of m1 and nj , for each j = 1, 2, 3. In fact, we just need to study the integral of g1 because k = 1. Moreover, studying the function g1 we observe that the only possibility to obtain an isolated zero of the function T f1 (X0 ) = g1 (X0 , t)dt 0

is assuming that n1 + n2 − m1 = 1. Otherwise, the only solution of f1 (X0 ) = 0 is X0 = 0, which corresponds to the equilibrium point (X0 , Y0 , Z0 ) = (0, 0, 1) of

1.2. Introduction: the classical theory

29

system (1.62). The same occurs if n2 and n3 are greater than 1 simultaneously. This analysis reduces the existence of possible periodic solutions to the following cases: (p1 ) n2 = 1 and n3 = 1; (p2 ) n2 > 1 and n3 = 1; (p3 ) n2 = 1 and n3 > 1. In the case (p1 ) we have Mz−1 (t)F1 (t, x(t, z, 0)) = −AX0 + BX0 + AU (t), and then f1 (X0 ) = (−A + B)T X0 + AK. Consequently, if A = B, then f1 (X0 ) = 0 implies X0 = AK/(T (A − B)). So, by Theorem 1.2.9, system (1.60) has a periodic solution (X(t, ε), Y (t, ε), Z(t, ε)) such that

AK AK , ,1 (X(0, ε), Y (0, ε), Z(0, ε)) −→ (X0 , Y0 , Z0 ) = T (A − B) T (A − B) when ε → 0. Note that the point (X0 , Y0 , Z0 ) is an equilibrium point of the system (1.60). Then, taking n1 = n2 = n3 = 1 and going back through the rescaling (1.56) of the variables and parameters, we obtain that the periodic solution of system (1.60) becomes the periodic solution (x(t), y(t), z(t)) of system (1.55) satisfying

aI aI , ,1 . (x(t), y(t), z(t)) ≈ T (a − b) T (a − b) Indeed, we observe that x0 = εX0 = ε

(aε−1 )(Iε−1 ) aI = . −1 T ε (a − b) T (a − b)

Moreover, we note that f1 (x0 ) = εf1 (X0 ) = −a + b = 0 so, the periodic orbit corresponding to x0 is stable if a > b, and unstable otherwise. This completes the proof of Theorem 1.2.13. Analogously the function f1 in the cases (p2 ) and (p3 ) is f1 (X0 ) = T BX0 + AK

and f1 (X0 ) = −T AX0 + AK,

respectively. In the ﬁrst case the condition f1 (X0 ) = 0 implies X0 = −(AK)/(T B). Now we observe that n2 > 1 and n3 = 1. So, going back through the rescaling, we obtain (−aε−n2 )(Iε−n1 ) aI x0 = εX0 = ε =− −1 n T bε T bε 1+n2 −2 and consequently, choosing n1 = 0 and n2 = 2, we get x0 = −aI/(T b). Note also that f1 (x0 ) = T b > 0, then the periodic orbit corresponding to x0 is always unstable. Thus, Theorem 1.2.14 is proved.

30

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Finally, in the case (p3 ), f1 (X0 ) = 0 implies X0 = K/T . So, taking n1 = 1 and going back through the rescaling, we have x0 = εX0 = εI/(T ε) = I/T . Additionally, f1 (x0 ) = −T a < 0. Therefore, the periodic solution coming from x0 is always stable. This proves Theorem 1.2.15. Proof of Theorem 1.2.16. As in the proofs of Theorems 1.2.13, 1.2.14 and 1.2.15, we start by considering a more general case in the powers of ε in (1.59), taking n2 > 0 and m2 < m1 < L. In this case the function F0 (t, x) of system (1.34) is ⎛ ⎞ 0 ⎠. F0 (t, x) = ⎝ Y (1.63) 1−Z Then the solution x(t, z, 0) of system (1.35) satisfying x(0, z, 0) = z is (X(t), Y (t), Z(t)) = (X0 , e−t Y0 , 1 − e−t + e−t Z0 ). This solution is periodic if Y0 = 0 and Z0 = 1. Then, through every point in the straight line Y = 0, Z = 1 there passes a periodic orbit lying in the phase space (X, Y, Z, t) ∈ R3 × S1 . We observe that using the notation of Subsection 1.2.5 we have n = 3, k = 1, α = X0 and β(α) = (0, 1). Consequently, M is a one dimensional manifold given by M = {(X0 , 0, 1) ∈ R3 : X0 ∈ R}. The fundamental matrix Mz (t) from (1.36) and satisfying Mz (0) = Id3 (with F0 given by (1.63)), and its inverse Mz−1 (t) are, respectively ⎞ ⎞ ⎛ ⎛ 1 0 0 1 0 0 0 ⎠ and Mz−1 (t) ⎝ 0 et 0 ⎠ . Mz (t) = ⎝ 0 e−t 0 0 e−t 0 0 et Since the matrix Mz−1 (0) − Mz−1 T has an 1 × 2 zero matrix in the upper right corner, and a 2 × 2 lower right corner matrix

1 − eT 0 Δ= , 0 1 − eT with det(Δ) = (1 − eT )2 = 0, we can apply the averaging theory described in Subsection 1.2.5. Again, using the notations introduced in the proofs of Theorems 1.2.13, 1.2.14 and 1.2.15, since k = 1 we will look only to the integral of the ﬁrst coordinate of F = (f1 , f2 , f3 ). In this case we have g1 (X0 , Y0 , Z0 , t) = −εn2 AX0 + ε−m1 +n1 +n2 AU (t). Comparing this function g1 with the same function obtained in the proof of Theorems 1.2.13, 1.2.14 and 1.2.15, it is easy to see that this case corresponds to the case (p3 ) of the mentioned theorems. Then, in order to have periodic solutions, we need to choose n2 = 1 and n1 + n2 − m1 = 1. So, following the steps of the proof of case (p3 ) by choosing n1 = 1 and coming back through the rescaling (1.56) to system (1.55), Theorem 1.2.16 is proved.

1.2. Introduction: the classical theory

31

Proof of Theorem 1.2.17. We start by considering the system (1.59) with n3 = 2, n2 > 0, m1 = n1 + n2 and m2 < m1 < m2 + n3 . With these conditions the system (1.59) becomes dX = −εn2 AX + εm2 −n1 −n2 +n3 BY + AU (t), dt dY = −Y + ε−m2 +n1 +n2 XZ, dt dZ = 1 − Z − εm2 +n1 +n2 XY. dt

(1.64)

Again, we will use the averaging theory described in Subsection 1.2.5. So, considering x = (X, Y, Z)T we obtain ⎛

⎞ AU (t) F0 (t, x) = ⎝ −Y ⎠ . 1−Z

(1.65)

Now we note that the solution x(t, z, 0) = (X(t), Y (t), Z(t)) such that x(0, z, 0) = z = (X0 , Y0 , Z0 ) of the system x˙ = F0 (t, x)

(1.66)

is

t

X(t) = X0 +

Y (t) = e−t Y0 ,

AU (s)ds,

Z(t) = 1 − e−t + e−t Z0 .

Since I = 0 and J(t) = 0 for 0 < t < T , in order that x(t, z, 0) is a periodic solution we need to ﬁx Y0 = 0 and Z0 = 1. This implies that through every point in a neighbourhood of X0 in the straight line Y = 0, Z = 1 there passes a periodic orbit lying in the phase space (X, Y, Z, t) ∈ R3 × S1 . Following the notation of Subsection 1.2.5, we have n = 3, k = 1, α = X0 and β(X0 ) = (0, 1). Hence, M is a one dimensional manifold M = {(X0 , 0, 1) ∈ R3 : X0 ∈ R}, and the fundamental matrix Mz (t) of (1.66) satisfying Mz (0) = Id3 is ⎛ ⎞ 1 0 0 ⎝ 0 e−t 0 ⎠. 0 0 e−t It is easy to see that the matrix Mz−1 (0) − Mz−1 (T ) has a 1 × 2 zero matrix in the upper right corner, and a 2 × 2 lower right corner matrix

Δ=

1 − eT 0

0 1 − eT

,

32

Chapter 1. The Averaging Theory for Computing Periodic Orbits

with det(Δ) = (1 − eT )2 = 0. Then, the hypotheses of Theorem 1.2.9 are satisﬁed. Now the components of the function Mz−1 (t)F (t, x(t, z, 0)) are

t n2 g1 (X0 , t) = −ε A X0 + AU (s)ds + AU (t), 0

t −m2 +n1 +n2 AU (s)ds et , X0 + g2 (X0 , t) = ε 0

g3 (X0 , t) = 0. Taking n1 = n2 = 1 and observing that k = 1 and n = 3, we are interested only in the ﬁrst component of the function F1 = (F11 , F12 , F13 ) described in Subsection 1.2.5. Indeed, applying the averaging theory, we must study the zeros of the ﬁrst component of the function T Mz−1 (t, z)F11 (t, x(t, z))dt. F (X0 ) = (f1 (X0 ), f2 (X0 ), f3 (X0 )) = 0

t F11 = −A X0 + AU (s)ds ,

Since

we deduce

T

f1 (X0 ) =

t −A X0 + AU (s)ds dt

0 T

= −AT X0 − A

2

t

U (s)ds ds. 0

Therefore, from f1 (X0 ) = 0 we obtain t A T U (s)ds ds = 0. X0 = − T 0 0 So, rescaling (1.56), we get x0 = ε X0 = −ε 2

2 aε

−1

εT

T

a J(s)ds = − T

T

J(s)ds. 0

Moreover, since f1 (x0 ) = −a/T < 0, because a and ε are positive, the T periodic orbit detected by the averaging theory is always stable. This ends the proof.

1.2.5 Another ﬁrst order averaging method for periodic orbits The next result proved in [56] extends the result of Theorem 1.2.9 to the case n = 2m and when the matrix Δα is the zero matrix. Here, ξ ⊥ : Rn = Rm × Rm → Rm is the projection of Rn onto its second set of m coordinates, i.e., ξ ⊥ (x1 , . . . , xn ) = (xm+1 , . . . , xn ).

1.2. Introduction: the classical theory

33

Theorem 1.2.18. Let V ⊂ Rm be open and bounded, let β0 : Cl(V ) → Rm be a C k function and Z = {zα = (α, β0 (α)) | α ∈ Cl(V )} ⊂ Ω its graphic in R2m . Assume that for each zα ∈ Z the solution x(t, zα ) of (1.34)ε=0 is T -periodic and that there exists a fundamental matrix Mzα (t) of (1.3) such that the matrix Mz−1 (0) − α (T ) has in the upper right corner the m× m matrix Ωα with det(Ωα ) = 0, and Mz−1 α in the lower right corner the m×m zero matrix. Consider the function G : Cl(V ) → Rm deﬁned by T ⊥ −1 G(α) = ξ Mzα (t)F1 (t, x(t, zα ))dt . (1.67) 0

If there is α0 ∈ V with G(α0 ) = 0 and det((∂G/∂α)(α0 )) = 0 then, for ε = 0 suﬃciently small, there is a unique T -periodic solution x(t, ε) of the system (1.34) such that x(t, ε) → x(t, zα0 ) as ε → 0. Theorem 1.2.18 is proved in Subsection 1.2.8. In the next subsubsection we provide some applications. A class of Duﬃng diﬀerential equations Many diﬀerent classes of Duﬃng diﬀerential equations have been studied by different authors. They are mainly interested in the existence of periodic solutions, in their multiplicity, stability, bifurcation, etc. See for instance the survey of J. Mawhin [70], and the articles [26, 73]. In this subsubsection we shall study the class of Duﬃng diﬀerential equations of the form x + cx + a(t)x + b(t)x3 = h(t), (1.68) where c > 0 is a constant, and a(t), b(t) and h(t) are continuous T -periodic functions. These diﬀerential equations were studied by Chen–Li in the papers [15, 16]. Their results were improved in [5] by Benterki–Llibre; we present a part of these improvements here, as an application of Theorem 1.2.18. Instead of working with the Duﬃng diﬀerential equation (1.68) we shall work with the equivalent diﬀerential system x = y, y = −cy − a(t)x − b(t)x3 + h(t). Theorem 1.2.19. For every simple real root of the polynomial T T 3 q(x0 ) = − b(s) ds x0 − a(s) ds x0 + 0

(1.69)

T

h(s) ds, 0

the diﬀerential system (1.69) has a periodic solution (x(t), y(t)) with (x(0), y(0)) close to (x0 , 0).

34

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Proof. We start by doing a rescaling of the variables (x, y), of the functions a(t), b(t) and h(t), and of the parameter c as follows x = εX, c = εC, b(t) = ε−1 B(t),

y = ε2 Y, a(t) = εA(t), h(t) = ε2 H(t).

(1.70)

Then the system (1.69) becomes X˙ = εY, Y˙ = −εCY − A(t)X − B(t)X 3 + H(t).

(1.71)

We shall apply the averaging Theorem 1.2.18 to system (1.71) and we shall obtain Theorem 1.2.19. In what follows, we shall use the notation from Theorem 1.2.18. Thus x = (X, Y )T and

0 , F0 (t, x) = −A(t)X − B(t)X 3 + H(t)

Y , F1 (t, x) = −CY

0 F2 (t, x) = . 0 The diﬀerential system (1.71) with ε = 0 has x(t, z, 0) = (X(t), Y (t))T as a solution with x(0, z, 0) = z = (X0 , Y0 )T , and where X(t) = X0 ,

Y (t) = Y0 + 0

t

−B(s)X03 − A(s)X0 + H(s) ds.

For x(t, z, 0) to be a periodic solution, X0 must satisfy 0

T

−B(s)X03 − A(s)X0 + H(s) ds = 0,

(1.72)

and Y0 is arbitrary. Therefore we get ¯0 , zα = (α, β0 (α)) = Y0 , X ¯ 0 is a real root of the cubic polynomial (1.72). In short, the unperturbed where X system (i.e., system (1.71) with ε = 0) has at most three families of periodic ¯ 0 is a real root of the cubic polynomial solutions because Y0 is arbitrary and X (1.72). Therefore, using the notation of Theorem 1.2.18, we have n = 2 and m = 1 for each one of these possible families of periodic solutions.

1.2. Introduction: the classical theory

35

We compute the fundamental matrix Mzα (t) associated to the ﬁrst varia˙ given by (1.71) with tional system (1.36), associated to the vector ﬁeld (Y˙ , X) ε = 0 and such that Mzα (0) = Id2 , and we obtain ⎛ ⎞ t 2 1 − 3B(s)X + A(s) ds 0 ⎠. Mzα (t) = ⎝ 0 0 1 The matrix

⎛

Mz−1 (0) − Mz−1 (T ) = ⎝ α α

−

T

⎞ 2 3B(s)X0 + A(s) ds ⎠ 0

¯ 0 of the has a non-zero 1 × 1 matrix in the upper right corner if the real root X cubic polynomial (1.72) is simple, and a zero 1 × 1 matrix in its lower right corner. Therefore, the assumptions of Theorem 1.2.18 hold and, by applying this theorem, we study the periodic solutions which can be prolonged from the unperturbed diﬀerential system to the perturbed one. Since for our diﬀerential system we have ξ ⊥ (Y, X) = X, we must compute the function G(α) = G(Y0 ) given in (1.4), i.e., T

G(Y0 ) = ξ ⊥

Mz−1 (t)F1 (t, x(t, zα , 0))dt α

T

=−

CY0 = −CT Y0 . 0

Theorem 1.2.18 says that, for every simple real root Y0 = 0 of the polynomial G(Y0 ), the diﬀerential system (1.71) with ε = 0 suﬃciently small has a periodic ¯ 0 , 0) when ε → 0, being solution (X(t), Y (t)) such that (X(0), Y (0)) tends to (X ¯ X0 a simple real root of the cubic polynomial (1.72). Now it is easy to check that the cubic polynomial (1.72) after the change of variables (1.70), i.e., X=

x , ε

Y =

y , ε2

H(t) =

h(t) , ε2

B(s) = εb(s),

A(s) =

becomes the polynomial q(x0 ). Hence the theorem is proved.

a(s) , ε

1.2.6 Proof of Theorem 1.2.1 Proof of Theorem 1.2.1(i). The assumptions guarantee the existence and uniqueness of the solutions of the initial valued problems (1.3) and (1.4) on the time-scale 1/ε. We introduce t [F (s, x) − f 0 (x)]ds. (1.73) u(t, x) = 0

Since we have subtracted the average of f (s, x) in the integrand, the integral is bounded, i.e., ||u(t, x)|| ≤ 2M T, t ≥ 0, x ∈ D.

36

Chapter 1. The Averaging Theory for Computing Periodic Orbits

We now introduce a transformation near the identity x(t) = z(t) + εu(t, z(t)).

(1.74)

This transformation will be used for simplifying equation (1.3). Diﬀerentiation of (1.74) and substitution in (1.3) yields x˙ = z˙ + ε

∂ ∂ u(t, z) + ε u(t, z)z˙ = εF (t, z + εu(t, z)) + ε2 R(t, z + εu(t, z), ε). ∂t ∂z

Using (1.73), we write this equation in the form

∂ I + ε u(t, z) z˙ = εf 0 (z) + S, ∂z with I the n × n identity matrix, and where S = εF (t, z + εu(t, z)) − εF (t, z) + ε2 R(t, z + εu(t, z), ε). Since ∂u/∂z is uniformly bounded (as u) we can invert to obtain

−1 ∂ ∂ I + ε u(t, z) = I − ε u(t, z) + O(ε2 ), ∂z ∂z

t ≥ 0,

z ∈ D.

(1.75)

From the Lipschitz continuity of F (t, z) we have ||F (t, z + εu(t, z)) − F (t, z)|| ≤ Lε||u(t, z)|| ≤ Lε2M T, where L is the Lispchitz constant. Due to the boundedness of R it follows that, for some positive constant C independent from ε, we have ||S|| ≤ ε2 C,

t ≥ 0,

z ∈ D.

(1.76)

From (1.75) and (1.76) we get that z˙ = εf 0 (z) + S − ε2

∂u 0 f (z) + O(ε3 ), ∂z

z(0) = x(0).

(1.77)

As S = O(ε2 ) by introducing the time-like variable τ = εt, we obtain that the solution of dy = f 0 (y), y(0) = z(0) dτ approximates the solution of (1.77) with error O(ε) on the time-scale 1 in τ , i.e., on the time-scale 1/ε in t. Due to the near identity transformation (1.74) we obtain that x(t) − y(t) = O(ε) (1.78) in the time-scale 1/ε.

1.2. Introduction: the classical theory

37

Now we shall impose the periodicity condition after which we can apply the Implicit Function Theorem. We transform x → z with the near identity transformation (1.74), then the equation for z becomes z˙ = εf 0 (z) + ε2 S(t, z, ε).

(1.79)

Due to the choice of u(t, z(t)), a T -periodic solution z(t) produces a T -periodic solution x(t). For S we have the expression S(t, z, ε) =

∂F ∂u (t, z)u(t, z) − (t, z)f 0 (z) + R(t, z, 0) + O(ε). ∂z ∂z

This expression is T -periodic in t and continuously diﬀerentiable with respect to z. Equation (1.79) is equivalent to the integral equation

t

z(t) = z(0) + ε

f (z(s))ds + ε 0

t

S(s, z(s), ε)ds.

2 0

The solution z(t) is T -periodic if z(t + T ) = z(t) for all t ≥ 0, which leads to the equation T T f 0 (z(s))ds + ε S(s, z(s), ε)ds = 0. (1.80) h(z(0), ε) = 0

Note that this is a short-hand notation. The right hand side of equation (1.80) does not depend on z(0) explicitly. But the solutions depend continuously on the initial values and so the dependence on z(0) is implicitly by the bijection z(0) → z(x). It is clear that h(p, 0) = 0. If ε is in a neighborhood of ε = 0, then equation (1.80) has a unique solution x(t, ε) = z(t, ε) because of the assumption on the Jacobian determinant (1.6). If ε → 0 then z(0, ε) → p. This completes the proof of statement (i). For proving statement (ii) of Theorem 1.2.1 we need some preliminary results. The ﬁrst result is Gronwall’s inequality. Lemma 1.2.20. Let a be a positive constant. Assume that t ∈ [t0 , t0 + a] and

t

ϕ(t) ≤ δ1

ψ(s)ϕ(s)ds + δ2 ,

(1.81)

t0

where ψ(t) ≤ 0 and ϕ(t) ≤ 0 are continuous functions, and δi > 0 for i = 1, 2. Then, t δ ψ(s)ds ϕ(t) ≤ δ2 e 1 t0 . Proof. From (1.81) we get

δ1

t t0

ϕ(t) ψ(s)ϕ(s)ds + δ2

≤ 1.

38

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Multiplying by δ1 ψ(t) and integrating we obtain

t

t0

therefore

δ1

δ1 ψ(s)ϕ(s) ds ≤ δ1 t0 ψ(r)ϕ(r)dr + δ2

s

t

ψ(s)ds, t0

t t ψ(s)ϕ(s)ds + δ2 − log δ2 ≤ δ1 ψ(s)ds. log δ1 t0

t0

Hence,

t

ψ(s)ϕ(s)ds + δ2 ≤ δ2 e

δ1

δ1

t

t0

ψ(s)ds

.

t0

From (1.81) the lemma follows. We consider the linear diﬀerential system x˙ = Ax,

(1.82)

where A is a constant n × n matrix. The eigenvalues λ1 , . . . , λn of system (1.82) are the zeros of the characteristic polynomial det(A − λId). If these eigenvalues λk are diﬀerent, with eigenvectors ek for k = 1, . . . , n, then ek eλk t , for k = 1, . . . , n, are n independent solutions of the system (1.82). Assume now that not all eigenvalues are diﬀerent, thus suppose that the eigenvalue λ has multiplicity m > 1. Then λ generates m independent solutions of the system (1.82) of the form P0 eλt , P1 (t)eλt , . . . , Pm−1 (t)eλt , where Pi (t) for i = 0, 1, . . . , m − 1 are polynomial vectors of degree at most i. With n independent solutions x1 (t), . . . , xn (t) of system (1.82) we form a matrix Φ(t) = (x1 (t), . . . , xn (t)), called a fundamental matrix of system (1.82). Every solution x(t) of system (1.82) can be written as x(t) = Φ(t)c, where c is a constant vector. Moreover the solution x(t) with x(t0 ) = x0 is x(t) = Φ(t)Φ(t0 )−1 x0 . (1.83) Usually, we choose the fundamental matrix Φ(t) in such a way that Φ(t0 ) = Id. From (1.83) and the explicit form of the independent solutions of system (1.82), the next result follows easily. Proposition 1.2.21. We consider the linear diﬀerential system x˙ = Ax, where A is a constant n×n matrix with eigenvalues λ1 , . . . , λn . Then the following statements hold:

1.2. Introduction: the classical theory

39

(i) if Reλk < 0 for k = 1, . . . , n then, for each solution x(t) with x(t0 ) = x0 , there exist two positive constants C and μ satisfying ||x(t)|| ≤ C||x0 ||e−μt

and

lim x(t) = 0;

t→∞

(ii) if Reλk ≤ 0 for k = 1, . . . , n and the eigenvalues with Reλk = 0 are diﬀerent, then the solution x(t) is bounded for t ≥ t0 ; more precisely, ||x(t)|| ≤ C||x0 ||

with C > 0;

(iii) if there exists an eigenvalue λk with Reλk > 0, then in each neighborhood of x = 0 there are solutions x(t) such that lim ||x(t)|| = ∞.

t→∞

Under the assumptions of statement (i) of Proposition 1.2.21, the solution x = 0 is called asymptotically stable. Under the assumptions of statement (ii), the solution x = 0 is called Liapunov stable. Finally, under the assumptions of statement (iii) the solution x = 0 is called unstable. The next result is also known as the Poincar´e–Liapunov Theorem. Theorem 1.2.22. Consider the diﬀerential system x˙ = Ax + B(t)x + f (t, x),

x(t0 ) = x0 ,

(1.84)

where t ∈ R, A is a constant n × n matrix having all its eigenvalues with negative real part, and B(t) is a continuous n × n matrix such that limt→∞ ||B(t)|| = 0. The function f (t, x) is continuous in t and x, and Lipschitz in x in a neighborhood of x = 0. If f (t, x) = 0 uniformly in t, lim ||x||→0 ||x|| then there exists positive constants C, t0 , δ and μ such that ||x0 || ≤ δ implies ||x(t)|| ≤ C||x0 ||e−μ(t−t0 )

for t ≥ t0 .

The solution x = 0 is asymptotically stable and the attraction is exponential in a δ-neighborhood of x = 0. Proof. By Proposition 1.2.21 we have an estimate for the fundamental matrix of the diﬀerential system ˙ = AΦ, Φ(t0 ) = Id. Φ Since all the eigenvalues of the matrix A have negative real part, there exist positive constants C and μ0 such that ||Φ(t)|| ≤ Ce−μ0 (t−t0 ) ,

for t ≥ t0 .

40

Chapter 1. The Averaging Theory for Computing Periodic Orbits

From the assumptions on f and B, for δ0 > 0 suﬃciently small there exist a constant b(δ0 ) such that if ||x|| ≤ δ0 then ||f (t, x)|| ≤ b(δ0 )||x||

for t ≥ t0 ,

and if t0 is suﬃciently large ||B(t)|| ≤ b(δ0 ),

for t ≥ t0 .

The existence and uniqueness Theorem states that in a neighborhood of x = 0 the solution of the initial problem (1.84), exists for t0 ≤ t ≤ t1 . It can be shown that this solution is deﬁned for all t ≥ t0 . We claim that the initial problem (1.84) is equivalent to the integral equation

t

x(t) = Φ(t)x0 +

Φ(t − s + t0 )[B(s)x(s) + f (s, x(s))]ds.

(1.85)

t0

Now we prove the claim. The fundamental matrix Φ(t) of the diﬀerential system x˙ = Ax can be written as Φ(t) = eA(t−t0 ) . We substitute x = Φ(t)z into the diﬀerential system (1.84) and obtain dΦ(t) z + Φ(t)z˙ = AΦ(t)z + B(t)Φ(t)z + f (t, Φ(t)z). dt Since dΦ(t)/dt = AΦ(t), we get z˙ = Φ(t)−1 B(t)Φ(t)z + Φ(t)−1 f (t, Φ(t)z). Integrating this expression between t0 and t and multiplying by Φ(t) we get the integral equation (1.85). So the claim is proved. Using the estimates for Φ, B and f we have t ||x(t)|| ≤ ||Φ(t)||||x0 || + [||Φ(t − s + t0 )||||B(s)||||x(s)|| + ||f (s, x(s))||] ds t0

≤ Ce

−μ0 (t−t0 )

t

||x0 || +

Ce−μ0 (t−s) 2b||x(s)||ds

t0

for t0 ≤ t ≤ t2 ≤ t1 . Therefore

t

eμ0 (t−t0 ) ||x(t)|| ≤ C||x0 || +

Ce−μ0 (s−t0 ) 2b||x(s)||ds,

t0

for t0 ≤ t ≤ t2 , where t2 is determined by the condition ||x|| ≤ δ0 . Using now Gronwall’s inequality (Lemma 1.2.20 with φ(s) = 2Cb) we obtain e−μ0 (s−t0 ) ||x(t)|| ≤ C||x0 ||e2Cb(t−t0 ) ,

1.2. Introduction: the classical theory

41

or ||x(t)|| ≤ C||x0 ||e(2Cb−μ0 )(t−t0 ) . If δ, and consequently b, are suﬃciently small we have that μ = μ0 − 2Cb is positive, and the inequality of the statement in the theorem follows for t ∈ [t0 , t2 ]. Finally, if we choose ||x0 || such that ||x0 || ≤ δ0 , then ||x(t)|| decreases, consequently the solution x = 0 is asymptotically stable and the attraction is exponential in a δ-neighborhood of x = 0. Now we shall consider linear diﬀerential systems of the form x˙ = A(t)x,

(1.86)

where A(t) is a continuous T -periodic n × n matrix, i.e., A(t + T ) = A(t) for all t ∈ R. For these systems we can deﬁne again a fundamental matrix putting in each column of this matrix an independent solution of the system (1.86). The next result usually called the Floquet Theorem says that the fundamental matrix of system (1.86) can be written as a product of a T -periodic matrix and a non-periodic matrix in general. Theorem 1.2.23. Consider the linear diﬀerential system (1.86) with A(t) a continuous T -periodic n × n matrix. Then each fundamental matrix Φ(t) of system (1.86) can be written as the product of two n × n matrices Φ(t) = P (t)eBt , where P (t) is T -periodic and B is a constant matrix. Proof. Since Φ(t) is a fundamental matrix of system (1.86), Φ(t + T ) is also a fundamental matrix. Indeed, deﬁne τ = t + T , then dx = A(τ − T )x = A(τ )x. dτ Therefore Φ(τ ) is also a fundamental matrix. The fundamental matrices Φ(t) and Φ(t+T ) are linearly dependent, i.e., there exists a non-singular matrix C such that Φ(t + T ) = Φ(t)C. Let B be a constant matrix such that C = eBT . We claim that the matrix Φ(t)e−Bt is T -periodic. Write Φ(t)e−Bt = P (t). Then, P (t + T ) = Φ(t + T )e−B(t+T ) = Φ(t)Ce−BT e−Bt = Φ(t)e−Bt = P (t). This completes the proof of the theorem.

Remark 1.2.24. The matrix C introduced in the proof of Theorem 1.2.23 is called the monodromy matrix of system (1.86). The eigenvalues ρk of the matrix C are called the characteristic multipliers. Each complex number λk such that ρk = eλk T is called a characteristic exponent. The characteristic multipliers are determined uniquely. We can choose the exponents λk so that they coincide with the eigenvalues of the matrix B.

42

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Proposition 1.2.25. Consider the diﬀerential system x˙ = A(t)x + f (t, x),

(1.87)

in Rn with A(t) a T -periodic continuous matrix, f (t, x) continuous in t ∈ R and in x in a neighborhood of x = 0. Assume that lim

||x||→0

f (t, x) =0 ||x||

uniformly in t.

If the real parts of the characteristic exponents of the linear periodic diﬀerential system y˙ = A(t)y (1.88) are negative, then the solution x = 0 of system (1.87) is asymptotically stable. Proof. By remark 1.2.24 and Theorem 1.2.23, we can use the change of variables x = M (t)z being M (t) the periodic fundamental matrix solution of the system (1.88). Then, the diﬀerential system (1.87) becomes z˙ = Bz + M (t)−1 f (t, M (t)z). The constant matrix B has all its eigenvalues with negative real part. The solution z of the previous system satisﬁes the assumptions of Theorem 1.2.22 from which the proposition follows. Proposition 1.2.26. Consider the diﬀerential system x˙ = Ax + B(t)x + f (t, x)

with t ≥ t0 ,

(1.89)

in Rn , where A is a constant n× n matrix having at least one eigenvalue with positive real part, and B(t) is a continuous n× n matrix such that limt→∞ ||B(t)|| = 0. The function f (t, x) is continuous in t and x, and Lipschitz in x in a neighborhood of x = 0. If f (t, x) = 0 uniformly in t, lim ||x||→0 ||x|| then the solution x = 0 is unstable. Proof. Doing the change of variables x = Sy, where S is a non-singular constant n × n matrix, the system (1.89) becomes y˙ = S −1 ASy + S −1 B(t)Sy + S −1 f (t, Sy).

(1.90)

While the solution x(t) is real, in general, the solution y(t) will be complex. The instability for the solution y = 0 of system (1.90) implies the instability for the solution x = 0 of system (1.89). We assume that S can be taken in such a way that S −1 AS is diagonal, otherwise the proof is similar, or see [21, Chapter 13.1].

1.2. Introduction: the classical theory

43

Assume now that Re(λi ) ≥ σ > 0 for i = 1, . . . , k, and that Re(λi ) ≤ 0 for i = k + 1, . . . , n. Let 2

R =

k

|yi |

2

2

and r =

i=1

n

|yi |2 .

i=k+1

From system (1.90) we shall compute the derivatives of R2 and r2 with respect to t. First, we have d|yi |2 d(yi y i ) = = y˙ i y i + yi y˙ i dt dt = 2Reλi |yi |2 + (S −1 B(t)Sy)y i + yi (S −1 B(t)Sy)i + (S −1 f (t, Sy)i yi + yi (S −1 f (t, Sy)i . We can choose ε > 0, δ0 and δ such that, for t ≥ t0 and ||y|| ≤ δ, we have |S −1 B(t)Sy|i ≤ ε|yi |,

|(S −1 f (t, Sy)i | ≤ ε|yi |.

Therefore, k n 1 d(R2 − r2 ) ≥ (Reλi − ε)|yi |2 − (Reλi + ε)|yi |2 . 2 dt i=1 i=k+1

Taking 0 < ε ≤ σ/2, we obtain Reλi − ε ≥ σ − ε ≥ ε for i = 1, . . . , k, and Reλi + ε ≥ ε for i = k + 1, . . . , n. Then, 1 d(R2 − r2 ) ≥ ε(R2 − r2 ) 2 dt

for t ≥ t0 and ||y|| ≤ δ.

(1.91)

Taking the initial conditions in such a way that (R2 − r2 )t=t0 = k > 0, from (1.91) we get that ||y||2 ≥ R2 − r2 ≥ ke2ε(t−t0 ) . Hence, this solution leaves the ball ||y|| ≤ δ. Consequently, y = 0 is unstable.

Proof of Theorem 1.2.1(ii). We linearize equation (1.3) in a neighborhood of the periodic solution x(t, ε). After translating x = z + x(t, ε), expanding with respect to z, omitting the non-linear terms and renaming the dependent variable again by x, we get the linear diﬀerential equation with T -periodic coeﬃcients x˙ = εA(t, ε)x, where A(t, ε) =

∂ [F (t, x) + εR(t, x, ε)]x=xε (t) . ∂x

(1.92)

44

Chapter 1. The Averaging Theory for Computing Periodic Orbits We deﬁne the T -periodic matrix B(t) =

∂F (t, p), ∂x

and from statement (i) we have limε→0 A(t, ε) = B(t). We also deﬁne the matrices 1 B = T

T

B(t)dt

T

[B(s) − B 0 ]ds.

and C(t) =

Note that B 0 is the matrix of the linearized averaging system. The matrix C(t) is T -periodic and its average is zero. The near-identity transformation x → y deﬁned by y = (I − εC(t))x provides ˙ y˙ = −εC(t)x + (I − εC(t))x˙ = −εB(t)x + εB 0 x + (I − εC(t))εA(t, ε)x = [εB 0 + ε(A(t, ε) − B(t)) − ε2 C(t)]A(t, ε)](I − εC(t))−1 y

(1.93)

= εB 0 y + ε(A(t, ε) − B(t))y + ε2 S(t, ε)y. The function S(t, ε) is T -periodic and bounded. We note that A(t, ε) − B(t) → 0 when ε → 0, and also that the characteristic exponents of the diﬀerential system (1.93) depend continuously on the small parameter ε. Therefore, for ε suﬃciently small, the sign of the real parts of the characteristic exponents is equal to the sign of the real parts of the eigenvalues of the matrix B 0 . The same conclusion holds, using the near-identity transformation, for the characteristic exponents of the diﬀerential system (1.92). Applying now Proposition 1.2.25 we obtain the stability of the periodic solution in the case of negative real parts. If at least one real part is positive, the Floquet transformation and the application of Proposition 1.2.26 provides the instability of the periodic solution.

1.2.7 Proof of Theorem 1.2.9 Proof of Theorem 1.2.9. We consider the function f : D × (−ε0 , ε0 ) → Rn , given by f (z, ε) = x(T, z, ε) − z. (1.94) Then, every (zε , ε) such that f (zε , ε) = 0 provides the periodic solution x(·, zε , ε) of (1.34). We need to study the zeros of the function (1.94), or, equivalently, of g(z, ε) = Y −1 (T, z)f (z, ε).

(1.95)

1.2. Introduction: the classical theory

45

We have that g (zα , 0) = 0, because x(·, zα , 0) is T -periodic, and we shall prove that dg Gα = (zα , 0) = Yα−1 (0) − Yα−1 (T ). (1.96) dz For this, we need to know (∂x/∂z) (·, z, 0). Since it is the matrix solution of (1.36) with (∂x/∂z) (0, z, 0) = In , we have that (∂x/∂z) (t, z, 0) = Y (t, z)Y −1 (0, z). Moreover, ∂x df (z, 0) = (T, z, 0) − In = Y (T, z)Y −1 (0, z) − In dz ∂z and

∂Y −1 dg ∂Y −1 −1 −1 (z, 0) = Y (0, z) − Y (T, z) + (T, z)f (z, 0), . . . , (T, z)f (z, 0) , dz ∂z1 ∂zn which, for zα ∈ Z, reduces to (1.96). We have ∂g ∂x (z, 0) = Y −1 (T, z) (T, z, 0). ∂ε ∂ε The function (∂x/∂ε) (·, z, 0) is the unique solution of the initial value problem y = Dx F0 (t, x(t, z, 0))y + F1 (t, x(t, z, 0)), Hence, ∂x (t, z, 0) = Y (t, z) ∂ε Now we have ∂g (z, 0) = ∂ε

T

t

y(0) = 0.

Y −1 (s, z)F1 (s, x(s, z, 0))ds.

Y −1 (s, z)F1 (s, x(s, z, 0))ds,

and hence

∂ (πg) (zα , 0) = f1 (α), ∂ε where f1 is given by (1.37). Applying Theorem 2.1, there exists αε ∈ V such that g(zαε , ε) = 0 and, further, f (zαε , ε) = 0, which assures that ϕ(·, ε) = x(·, zαε , ε) is a T -periodic solution of (1.34).

1.2.8 Proof of Theorem 1.2.18 Since the result of Theorem 1.2.18 is analogous to the result of Theorem 1.2.9, their proofs are similar. Proof of Theorem 1.2.18. Since Z is a compact set and x(t, zα ) is T -periodic for each zα ∈ Z, there is an open neighborhood D of Z in Ω, and 0 < ε1 ≤ ε0 such that any solution x(t, z, ε) of (1.34) with initial conditions in D×(−ε1 , ε1 ) is well deﬁned in [0, T ]. We consider the function L : D × (−ε1 , ε1 ) → R2m , (z, ε) → x(T, z, ε) − z. If (¯ z, ε¯) ∈ D × (−ε1 , ε1 ) is such that L(¯z, ε¯) = 0, then x(t, z¯, ε¯) is a T -periodic

46

Chapter 1. The Averaging Theory for Computing Periodic Orbits

solution of (1.34)ε=¯ε . Clearly, the converse is also true. Hence, the problem of ﬁnding T -periodic orbits of (1.34) close to the periodic orbits with initial conditions in Z is reduced to ﬁnding the zeros of L(x, ε). ε) = Mz−1 (T )L(z, ε) coincide, since The sets of zeros of L(z, ε) and L(z, Mz (T ) is a fundamental matrix. Moreover, following the proof of Theorem 1.2.9, we can compute that T −1 ε) = M (0)−M −1(T ) +Dz Mz−1 (t)F1 (t, x(t, z, 0))dt ε+O(ε2 ). Dz L(z, z z 0

(1.97) −1 (0) ∩ (ξ ◦ L) −1 (0). From (1.97) we obtain −1 (0) = (ξ ⊥ ◦ L) We note that L α , 0) = Mz−1 (0)−Mz−1 (T ). If we write z ∈ R2m as z = (u, v) with u, v ∈ Rm , Dz L(z α α α , 0) is the upper right corner of Mz−1 (0)−Mz−1 (T ). Then, from (i), then Dv (ξ ◦ L)(z we can apply the Implicit Function Theorem, deducing the existence of an open neighborhood U × (−ε2 , ε2 ) of Cl(V ) in ξ(D) × (−ε1 , ε1 ), an open neighborhood O of β0 (Cl(V )) in Rm and a unique C k function β(α, ε) : U × (−ε2 , ε2 ) → O such −1 (0) ∩ (U × O × (−ε2 , ε2 )) is exactly the graphic of β(α, ε). Now, if we that (ξ ◦ L) β(α, ε), ε), then δ deﬁne the function δ : U × (−ε2 , ε2 ) → R as δ(α, ε) = (ξ ⊥ ◦ L)(α, k −1 is a function of class C and L (0)∩(U ×O ×(−ε2, ε2 )) = {(α, β(α, ε), ε) | (α, ε) ∈ −1 (0) in an open neighborhood of Z in δ −1 (0)}. Therefore, to describe the set L n −1 R × (−ε0 , ε0 ), it suﬃces to describe δ (0) in an open neighborhood of Cl(V ) in R × (−ε0 , ε0 ). (0) − Mz−1 (T ) has in the lower right corner the m × m zero matrix Since Mz−1 α α and δ(α, 0) = 0 in V × (−ε2 , ε2 ), the function δ(α, ε) can be written as δ(α, ε) = ε) in V × (−ε2 , ε2 ), where G(α) is the function given in (1.67), see εG(α) + ε2 G(α, ε) = G(α) + εG(α, ε) then δ −1 (0) = δ−1 (0). [12]. In addition, if δ(α, If there is α0 ∈ V such that δ(α0 , 0) = G(α0 ) = 0 and det((∂G/∂α)(α0 )) = 0 then, from the Implicit Function Theorem, there exist ε3 > 0 small, an open neighborhood V0 of α0 in V and a unique function α(ε) : (−ε3 , ε3 ) → V0 of class C k such that δ−1 (0) ∩ (V0 × (−ε3 , ε3 )) is the graphic of α(ε), which also represents the set δ −1 (0) ∩ (V0 × (−ε3 , ε3 )). This completes the proof of the theorem.

1.3 Averaging theory of arbitrary order and dimension for ﬁnding periodic solutions In this section we shall study periodic solutions of systems of the form x (t) =

k

εi Fi (t, x) + εk+1 R(t, x, ε),

(1.98)

i=0

where Fi : R × D → Rn for i = 0, 1, . . . , k, and R : R × D × (−ε0 , ε0 ) → Rn are locally Lipschitz functions, being T -periodic in the ﬁrst variable, and where D is

1.3. Averaging theory for arbitrary order and dimension

47

an open subset of Rn ; eventually F0 can be the zero constant function. The classical works using the averaging theory for studying the periodic solutions of a diﬀerential system (1.98) usually only provide this theory up to ﬁrst (k = 1) or second order (k = 2) in the small parameter ε. Moreover, these theories assume diﬀerentiability of the functions Fi and R up to class C 2 or C 3 , respectively. Recently, in [14], this averaging theory for computing periodic solutions was developed up to second order in dimension n, and up to third order (k = 3) in dimension 1, only using that the functions Fi and R are locally Lipschitz. Also, in the recent work [37], the averaging theory for computing periodic solutions was developed to an arbitrary order k in ε for analytical diﬀerential equations in dimension 1. In this section we shall develop the averaging theory for studying the periodic solutions of a diﬀerential system (1.98) up to arbitrary order k in dimension n, with zero or non-zero F0 , and where the functions Fi and R are only locally Lipschitz. In fact this section is based in the results of the paper Llibre–Novaes–Teixeira [55]. An example that qualitative new phenomena can be found only when considering higher order analysis is the following. Consider arbitrary polynomial perturbations x˙ = −y + j≥1 εj fj (x, y), (1.99) y˙ = x + j≥1 εj gj (x, y), of the harmonic oscillator, where ε is a small parameter. In this diﬀerential system the polynomials fj and gj are of degree n in the variables x and y, and the system is analytic in the variables x, y and ε. Then in [37] (see also Iliev [45]) it is proved that system (1.99) for ε = 0 suﬃciently small has no more than [s(n−1)/2] periodic solutions bifurcating from the periodic solutions of the linear center x˙ = −y, y˙ = x, using the averaging theory up to order s, and this bound can be reached. Here, [·] denotes the integer part function. So, higher order averaging theory can improve the results on the periodic solutions, both qualitatively and quantitatively. In short, the goal of this section is to extend the averaging theory for computing periodic solutions of the diﬀerential system in n variables (1.98) up to an arbitrary order k in ε for locally Lipschitz diﬀerential systems, using the Brouwer degree.

1.3.1 Statement of the main results We are interested in studying the existence of periodic orbits of general diﬀerential systems expressed by x (t) =

k

εi Fi (t, x) + εk+1 R(t, x, ε),

(1.100)

i=0

where Fi : R × D → Rn for i = 1, 2, . . ., k, and R : R × D × (−ε0 , ε0 ) → Rn are continuous functions, being T -periodic in the ﬁrst variable, and where D is an open subset of Rn .

48

Chapter 1. The Averaging Theory for Computing Periodic Orbits

In order to state our main results we introduce some notation. Let L be a positive integer, let x = (x1 , . . . , xn ) ∈ D, t ∈ R and yj = (yj1 , . . . , yjn ) ∈ Rn for j = 1, . . . , L. Given F : R × D → Rn a suﬃciently smooth function, for each (t, x) ∈ R × D we denote by ∂ L F (t, x) a symmetric L-multilinear map which is L applied to a “product” of L vectors of Rn , which we denote as j=1 yj ∈ RnL . The deﬁnition of this L-multilinear map is ∂ L F (t, x)

L

n

yj =

j=1

i1 ,...,iL

∂ L F (t, x) y1i · · · yLiL . ∂xi1 · · · ∂xiL 1 =1

(1.101)

We deﬁne ∂ 0 as the identity functional. Given a positive integer b and a vector y ∈ Rn , we also write y b = bi=1 y ∈ Rnb . Remark 1.3.1. The L-multilinear map deﬁned in (1.101) is the Lth Fr´echet derivative of the function F (t, x) with respect to the variable x. Indeed, for every ﬁxed t ∈ R, if we consider the function Ft : D → Rn such that Ft (x) = F (t, x), then (L) ∂ L F (t, x) = Ft (x) = ∂ L /∂xL F (t, x). Example 1.3.2. To illustrate the above notation (1.101), we consider a smooth function F : R × R2 → R2 . So, for x = (x1 , x2 ) and y 1 = (y11 , y21 ), we have ∂F (t, x)y 1 =

∂F ∂F (t, x)y11 + (t, x)y21 . ∂x1 ∂x2

Now, for y 1 = (y11 , y21 ) and y 2 = (y12 , y22 ), we have ∂ 2 F (t, x)(y 1 , y 2 ) =

∂ 2 F (t, x) 1 2 ∂ 2 F (t, x) 1 2 y y + y y ∂x1 ∂x1 1 1 ∂x1 ∂x2 1 2 ∂ 2 F (t, x) 1 2 ∂ 2 F (t, x) 1 2 y y + y y . + ∂x2 ∂x1 2 1 ∂x2 ∂x2 2 2

Observe that, for each (t, x) ∈ R × D, ∂F (t, x) is a linear map in R2 and ∂ 2 F (t, x) is a bilinear map in R2 × R2 . Let ϕ(·, z) : [0, tz ] → Rn be the solution of the unperturbed system x (t) = F0 (t, x)

(1.102)

such that ϕ(0, z) = z. For i = 1, 2, . . . , k, we deﬁne the averaged function of order i, fi : D → Rn , as yi (T, z) , (1.103) fi (z) = i! where yi : R × D → Rn , for i = 1, 2, . . . , k − 1, are deﬁned recurrently by the

1.3. Averaging theory for arbitrary order and dimension

49

integral equation yi (t, z) = i!

t Fi s, ϕ(s, z) 0

+

i l=1 Sl

1 b1 ! b2 !2!b2 · · · bl !l!bl

l ∂ L Fi−l s, ϕ(s, z) yj (s, z)bj ds, (1.104) j=1

where Sl is the set of all l-tuples of non-negative integers (b1 , b2 , . . . , bl ) satisfying b1 + 2b2 + · · · + lbl = l, and L = b1 + b2 + · · · + bl . In Subsection 1.3.3 we compute the sets Sl for l = 1, 2, 3, 4, 5. Furthermore, we make the functions fk (z) explicit, up to k = 5 when F0 = 0, and up to k = 4 when F0 = 0. Related to the averaging functions (1.103) there exist two cases of (1.100), essentially diﬀerent, that must be treated separately, namely, when F0 = 0 and when F0 = 0. It can be seen in the following remarks. Remark 1.3.3. If F0 = 0, then ϕ(t, z) = z for each t ∈ R. So, y1 (t, z) =

t

F1 (t, z)ds,

T

and f1 (t, z) =

F1 (t, z)dt, 0

as usual in averaging theory; see, for instance, [5]. Remark 1.3.4. If F0 = 0, then t y1 (t, z) = F1 (s, ϕ(s, z)) + ∂F0 (s, ϕ(s, z)) y1 (s, z)ds.

(1.105)

The integral equation (1.105) is equivalent to the following Cauchy problem: u(t) ˙ = F1 (t, ϕ(t, z)) + ∂F0 (t, ϕ(t, z)) u and u(0) = 0,

(1.106)

i.e., y1 (t, z) = u(t). If we write

t

η(t, z) =

∂F0 (s, ϕ(s, z))ds,

(1.107)

we have

y1 (t, z) = eη(t,z)

t

e−η(s,z) F1 (s, ϕ(s, z))ds

(1.108)

and

f1 (z) =

T

e−η(t,z) F1 (t, ϕ(t, z))dt.

Moreover, each yi (t, z) is obtained similarly from a Cauchy problem. The formulae are given explicitly in Subsection 1.3.3.

50

Chapter 1. The Averaging Theory for Computing Periodic Orbits

In the following, we state our main results: Theorem 1.3.5 when F0 = 0, and Theorem 1.3.6 when F0 = 0. The Brouwer degree dB , which is deﬁned in Appendix 1.3.6, is used. Theorem 1.3.5. Suppose that F0 = 0. In addition, for the functions of (1.100), we assume the following conditions: (i) for each t ∈ R, Fi (t, ·) ∈ C k−i for i = 1, 2, . . . , k; ∂ k−i Fi is locally Lipschitz in the second variable for i = 1, 2, . . . , k; and R is continuous and locally Lipschitz in the second variable; (ii) fi = 0 for i = 1, 2, . . . , r − 1 and fr = 0, where r ∈ {1, 2, . . . , k} (here, we are taking f0 = 0). Moreover, suppose that for some a ∈ D with fr (a) = 0, there exists a neighborhood V ⊂ D of a such that fr (z) = 0 for all z ∈ V \ {a}, and that dB (fr (z), V, a) = 0. Then, for |ε| > 0 suﬃciently small, there exists a T -periodic solution x(·, ε) of (1.100) such that x(0, ε) → a when ε → 0. Theorem 1.3.6. Suppose that F0 = 0. In addition, for the functions of (1.100), we assume the following conditions: (i) there exists an open subset W of D such that, for any z ∈ W , ϕ(t, z) is T -periodic in the variable t; (ii) for each t ∈ R, Fi (t, ·) ∈ C k−i for i = 0, 1, 2, . . . , k; ∂ k−i Fi is locally Lipschitz in the second variable for i = 0, 1, 2, . . . , k; and R is continuous and locally Lipschitz in the second variable; (iii) fi = 0 for i = 1, 2, . . . , r − 1 and fr = 0, where r ∈ {1, 2, . . . , k}; moreover, suppose that for some a ∈ W with fr (a) = 0, there exists a neighborhood V ⊂ W of a such that fr (z) = 0 for all z ∈ V \ {a}, and that dB (fr (z), V, a) = 0. Then, for |ε| > 0 suﬃciently small, there exists a T -periodic solution x(·, ε) of (1.100) such that x(0, ε) → a when ε → 0. Theorems 1.3.5 and 1.3.6 are proved in Subsection 1.3.2. Remark 1.3.7. When the functions fi deﬁned in (1.103), for i = 1, 2, . . . , k, are C 1 , the hypotheses (ii) from Theorem 1.3.5 and (iii) from Theorem 1.3.6 become: (iv) fi = 0 for i = 1, 2 . . . , r − 1 and fr = 0, where r ∈ {1, 2, . . . , k}; moreover, suppose that for some a ∈ W with fr (a) = 0 we have that fr (a) = 0. In this case, instead of Brouwer degree theory, the Implicit Function Theorem could be used to prove Theorems 1.3.5 and 1.3.6. We emphasize that our main contribution to the advanced averaging theory is based on Theorems 1.3.5 and 1.3.6. In fact, we provide conditions on the regularity of the functions, weaker than those given in [37].

1.3. Averaging theory for arbitrary order and dimension

51

1.3.2 Proofs of Theorems 1.3.5 and 1.3.6 Let g : (−ε0 , ε0 ) → Rn be a function deﬁned on a small interval (−ε0 , ε0 ). We say that g(ε) = O(ε ) for some positive integer if there exists constants ε1 > 0 and M > 0 such that ||g(ε)|| ≤ M |ε | for −ε1 < ε < ε1 . The symbol O is one of the Landau’s symbols (see for instance [78]). To prove Theorems 1.3.5 and 1.3.6 we need the following lemma. Lemma 1.3.8 (Fundamental Lemma). Under the assumptions of Theorems 1.3.5 or 1.3.6 let x(·, z, ε) : [0, tz ] → Rn be the solution of (1.100) with x(0, z, ε) = z. If tz = T , then k yi (t, z) + εk+1 O(1), εi x(t, z, ε) = ϕ(t, z) + i! i=1 where yi (t, z) are deﬁned in (1.104), for i = 1, 2, . . . , k. Proof. By continuity of the solution x(t, z, ε) and by compactness of the set [0, T ]× V × [−ε1 , ε1 ], there exits a compact subset K of D such that x(t, z, ε) ∈ K for all t ∈ [0, T ], z ∈ V and ε ∈ [−ε1 , ε1 ]. Now, by continuity of the function R, |R(s, x(s, z, ε), ε)| ≤ max{|R(t, x, ε)|, (t, x, ε) ∈ [0, T ] × K × [−ε1 , ε1 ]} = N . Then t R(s, x(s, z, ε), ε)ds ≤ 0

T

|R(s, x(s, z, ε), ε)| ds = T N,

which implies that

t

R(s, x(s, z, ε), ε)ds = O(1).

(1.109)

Related to the functions x(t, z, ε) and ϕ(t, z) we have the following two equalities: x(t, z, ε) = z +

k i=0 t

ϕ(t, z) = z +

t

Fi (s, x(s, z, ε))ds + O(εk+1 ),

εi 0

(1.110)

F0 (s, ϕ(s, z))ds. 0

Moreover, x(t, z, ε) = ϕ(t, z) + O(ε). Indeed, F0 is locally Lipschitz in the second variable, so from the compactness of the set [0, T ] × V × [−ε0 , ε0 ] and from (1.110) it follows that t |x(t, z, ε) − ϕ(t, z)| ≤ |F0 (s, x(s, z, ε)) − F0 (s, ϕ(s, z))|ds 0 t |F1 (s, x(s, z, ε))|ds + O(ε2 ) + |ε| 0 t L0 |x(s, z, ε) − ϕ(s, z)|ds < |ε|M eT L0 . ≤ |ε|M + 0

52

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Here, L0 is the Lipschitz constant of F0 on the compact K. The ﬁrst and second inequalities were obtained similarly to (1.109). The last inequality is a consequence of the Gronwall Lemma (see, for example, [78, Lemma 1.3.1]). In order to prove the present lemma we need the following claim. Claim. For some positive integer m, let G : R × D → Rn be a C m function. Then G(t, x(t, z, ε)) 1 = λm−1 1 0

+

m

1 ∂ m G t, m ◦ m−1 ◦ · · · ◦ 1 (x(t, z, ε)) λm−1 0 0 0 m − ∂ G(t, ϕ(t, z)) dλm dλm−1 · · · dλ1 · (x(t, z, ε) − ϕ(t, z))m 1

λm−2 ··· 2

∂ L G(t, ϕ(t, z))

L=0

1

(x(t, z, ε) − ϕ(t, z))L , L!

where i (v) = λi v + (1 − λi )ϕ(t, z) for v ∈ Rn . We shall using induction on m. For m = 1, G ∈ C 1 . Let prove this claim 1 (λ1 ) = G t, 1 (x(t, z, ε)) . So, 1 G(t, x(t, z, ε)) =G(t, ϕ(t, z)) + 1 (1) − 1 (0) = G(t, ϕ(t, z)) + 1 (λ1 )dλ1 =G(t, ϕ(t, z)) +

1

∂G(t, 1 (x(t, z, ε)))dλ1 · (x(t, z, ε) − ϕ(t, z))

1

=

∂G(t, 1 (x(t, z, ε))) − ∂G(t, ϕ(t, z)) dλ1 · (x(t, z, ε) − ϕ(t, z))

G(t, ϕ(t, z)) + ∂G(t, ϕ(t, z))(x(t, z, ε) − ϕ(t, z)). Given an integer k > 1, we assume as the inductive hypothesis (I1) that the claim is true for m = k − 1. Now, for m = k, G ∈ C k ⊂ C k−1 . So, from inductive hypothesis (I1), 1 1 1 1 k−3 G(t, x(t, z, ε)) = ∂ k−1 G t, k−1 ◦ k−2 ◦ · · · λk−2 λ · · · λ 1 2 k−2 0 0 0 0 k−1 G(t, ϕ(t, z)) dλk−1 dλk−2 · · · dλ1 ◦ 1 (x(t, z, ε)) − ∂ · (x(t, z, ε) − ϕ(t, z))k−1 +

k−1

∂ L G(t, ϕ(t, z))

L=0

(x(t, z, ε) − ϕ(t, z))L . L! (1.111)

Let (λk ) = ∂ k−1 G t, k ◦ k−1 ◦ · · · ◦ 1 (x(t, z, ε)) . So, 1

(λk )dλk = (1) − (0) = ∂ k−1 G t, k−1 ◦ k−2 ◦ · · · ◦ 1 (x(t, z, ε)) − ∂ m G(t, ϕ(t, z)). (1.112)

1.3. Averaging theory for arbitrary order and dimension

53

The derivative of (λk ) can be easily obtained as (λk ) = λk−1 λk−2 · · · λ1 ∂ k G t, k ◦ k−1 ◦ · · · ◦ 1 (x(t, z, ε)) (x(t, z, ε) − ϕ(t, z)). Hence, 1 0

(λk )dλk = λk−1 λk−2 · · · λ1

1

∂ k G t, k ◦ k−1 ◦ · · · ◦ 1 (x(t, z, ε))

0 − ∂ G(t, ϕ(t, z)) dλk · (x(t, z, ε) − ϕ(t, z)) k

+ λk−1 λk−2 · · · λ1 ∂ k G(t, ϕ(t, z))(x(t, z, ε) − ϕ(t, z)). (1.113) Therefore, from (1.111) and (1.113), we conclude that G(t, x(t, z, ε)) 1 1 1 1 k−2 λk−1 = ∂ k G t, k ◦ k−1 ◦ · · · ◦ 1 (x(t, z, ε)) λ · · · λ 1 2 k−1 0 0 0 0 k − ∂ G(t, ϕ(t, z)) dλk dλk−1 · · · dλ1 · (x(t, z, ε) − ϕ(t, z))k +

k

∂ L G(t, ϕ(t, z))

L=0

(x(t, z, ε) − ϕ(t, z))L . L!

This completes the proof of the claim. Given a non-negative integer m, we note that for a C m function G such that m ∂ G is locally Lipschitz in the second variable, the claim implies the equality m

(x(t, z, ε) − ϕ(t, z))L + O(εm+1 ). L! (1.114) Indeed, for m = 0, G is a continuous function locally Lipschitz in the second variable so, G(t, x(t, z, ε)) =

L L=0 ∂ G(t, ϕ(t, z))

|G(t, x(t, z, ε)) − G(t, ϕ(t, z))| ≤ LG |x(t, z, ε) − ϕ(t, z)| < |ε|LG M eT L0 . Here, LG is the Lipschitz constant of the function G on the compact K. Thus, G(t, x(t, z, ε)) = G(t, ϕ(t, z)) + O(ε). Moreover, for m ≥ 1 the claim implies (1.114) in a similar way to (1.109). Again, we shall use induction, now on k, to prove the present lemma. For k = 1, F0 ∈ C 1 and the functions ∂F0 and F1 are locally Lipschitz in the second variable. Thus, from (1.114) and taking G = F0 and G = F1 , we obtain F0 (t, x(t, z, ε)) = F0 (t, ϕ(t, z)) + ∂F0 (t, ϕ(t, z))(x(t, z, ε) − ϕ(t, z)) + O(ε2 ), F1 (t, x(t, z, ε)) = F1 (t, ϕ(t, z)) + O(ε), (1.115)

54

Chapter 1. The Averaging Theory for Computing Periodic Orbits

respectively. From (1.110) and (1.115) we compute d (x(t, z, ε) − ϕ(t, z)) = ∂F0 (t, ϕ(t, z)) (x(t, z, ε) − ϕ(t, z))+εF1 (t, ϕ(t, z))+O(ε2 ). dt (1.116) Solving the linear diﬀerential equation (1.115) with respect to x(t, z, ε) − ϕ(t, z) for the initial condition x(0, z, ε) − ϕ(0, z, ε) = 0, and comparing the solution with (1.108), we conclude that x(t, z, ε) = ϕ(t, z) + εy1 (t, z) + O(ε2 ). Given an integer k we assume as the inductive hypothesis (I2) that the lemma is true for k = k − 1. Now for k = k, Fi = C k−i for i = 0, 1, . . . , k and ∂ k−i Fi is locally Lipschitz in the second variable for i = 0, 1, . . . , k. So, from (1.114), Fi (t, x(t, z, ε)) =

k−i

∂ L Fi (t, ϕ(t, z))

L=0

(x(t, z, ε) − ϕ(t, z))L + O(εk−i+1 ), (1.117) L!

for i = 0, 1, . . . , k. Applying the inductive hypothesis (I2) in (1.117) we get Fi (t, x(t, z, ε)) = F1 (t, ϕ(t, z)) +

k−i

⎛

∂ L Fi (t, ϕ(t, z)) ⎝

k−i−L+1 i=1

L=1

⎞L y (t, z) i ⎠ + O(εk−i+1 ), εi i! (1.118)

for i = 1, 2, . . . , k. Now using the Multinomial Theorem in (1.118) (see, for instance, [43, pag. 186]), we obtain Fi (t, x(t, z, ε)) = Fi (t, ϕ(t, z)) +

k−i k−i L=1 l=L S k−1 l,L

εl b1 ! b2 !2!b2 · · · bk−1 !(k − 1)!bk−1

∂ L Fi (t, ϕ(t, z))

k−1

yj (t, z)bj

j=1

+ O(εk−i+1 ), n is the set of all n-tuples of non-negative integers for i = 1, 2, . . . , k. Here, Sl,L (b1 , b2 , . . . , bn ) satisfying b1 + 2b2 + · · · + nbn = l and b1 + b2 + · · · + bn = L. We note that if n > l then bl+1 = bl+2 = · · · = bn = 0. Hence,

Fi (t, x(t, z, ε)) = Fi (t, ϕ(t, z)) +

k−i k−i l L=1 l=L Sl,L

+ O(εk−i+1 ),

l εl L ∂ F (t, ϕ(t, z)) yj (t, z)bj i b1 ! b2 !2!b2 · · · bl !l!bl j=1

(1.119)

1.3. Averaging theory for arbitrary order and dimension

55

for i = 1, 2, . . . , k, because k − i ≥ l. l Finally, doing a change of indexes in (1.119), and observing that ∪lL=1 Sl,L = Sl , we may write Fi (t, x(t, z, ε)) = Fi (t, ϕ(t, z)) +

k−i l=1

εl

Sl

1 b1 ! b2 !2!b2 · · · bl !l!bl

∂ L Fi (t, ϕ(t, z))

l

yj (t, z)bj

(1.120)

j=1

+ O(εk−i+1 ), for i = 1, 2, . . . , k. Following the above steps we also obtain F0 (t, x(t, z, ε)) = F0 (t, ϕ(t, z)) + ∂F0 (t, ϕ(t, z))(x(t, z, ε) − ϕ(t, z)) k i 1 i L ε yj (t, z)bj + ∂ F (t, ϕ(t, z)) 0 b2 · · · b !i!br b ! b !2! 1 2 i i=1 j=1 Si

− ∂F0 (t, ϕ(t, z))

(1.121)

yi (t, z) + O(εk+1 ). i!

Now, from (1.110), we compute d (x(t, z, ε) − ϕ(t, z)) = F0 (t, x(t, z, ε)) dt k

− F0 (t, ϕ(t, z)) +

εi Fi (t, x(t, z, ε)) + O(εk+1 ).

i=1

(1.122) Proceeding with a change of index we obtain from (1.120) that k i=1

εi Fi (t, x(t, z, ε)) =

k i=1 l

εi

i−1 l=0 Sl

1 b1 ! b2

!2!b2

yj (t, z) + O(ε bj

· · · bl !l!bl

∂ L Fi−l (t, ϕ(t, z)) (1.123)

k+1

).

j=1

Substituting (1.121) and (1.123) in (1.122), we conclude that d (x(t, z, ε) − ϕ(t, z)) = ∂F0 (t, ϕ(t, z)) (x(t, z, ε) − ϕ(t, z)) dt ! i k 1 i ε ∂ L Fi−l (t, ϕ(t, z)) + b2 · · · b !l!bl b ! b !2! 1 2 l i=1 l=0 Sl " l y (t, z) i + O(εk+1 ). yj (s, z)bj − ∂F0 (t, ϕ(t, z)) i! j=1 (1.124)

56

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Solving the linear diﬀerential equation (1.124) with respect to x(t, z, ε) − ϕ(t, z) for the initial condition x(0, z, ε) − ϕ(0, z) = 0, we obtain x(t, z, ε) = ϕ(t, z) +

k

εi

i=1

where Yi (t, z) = e

!

t

η(t,z)

e

−η(s,z)

0 l

i l=0 Sl

Yi (t, z) + O(εk+1 ), i!

i! ∂ L Fi−l (s, ϕ(s, z)) b1 ! b2 !2!b2 · · · bl !l!bl "

yj (s, z)bj − ∂F0 (s, ϕ(s, z))yi (s, z) ds.

j=1

The function η(t, z) was deﬁned in (1.107). Hence, d Yi (t, z) = ∂F0 (t, ϕ(t, z))Yi (t, z) dt i l i! L + ∂ Fi−l (t, ϕ(t, z)) yj (t, z)bj b1 ! b2 !2!b2 · · · bl !l!bl j=1 l=0 Sl

− ∂F0 (t, ϕ(t, z))yi (t, z)ds. Computing the derivative of the function yi (t, z), we conclude that the functions yi (t, z) and Yi (t, z) are deﬁned by the same diﬀerential equation. Since Yi (0, z) = yi (0, z) = 0, it follows that Yr (t, z) ≡ yr (t, z) for every i = 1, 2, . . . , k. So, we have concluded the induction which completes the proof of the lemma. In a few words, the proof of Theorem 1.3.5 is an application of the Brouwer degree (see Appendix 1.3.6) to the approximated solution given by Lemma 1.3.8. Proof of Theorem 1.3.5. Let x(·, z, ε) be a solution of (1.100) such that x(0, z, ε) = z. For each z ∈ V , there exists ε1 > 0 such that if ε ∈ [−ε1 , ε1 ] then x(·, z, ε) is deﬁned in [0, T ]. Indeed, by the Existence and Uniqueness Theorem of solutions (see, for example, [78, Theorem 1.2.4]), x(·, z, ε) is deﬁned for all 0 ≤ t ≤ inf (T, d/M (ε)), where k i k+1 ε Fi (t, x) + ε R(t, x, ε) M (ε) ≥ i=1

for all t ∈ [0, T ], for each x with |x − z| < d, and for every z ∈ V . When ε is suﬃciently small we can take d/M (ε) suﬃciently large in order that inf (T, d/M (ε)) = T for all z ∈ V . We write εf (z, ε) = x(T, z, ε)−z. From Lemma 1.3.8 and equation (1.109) we have that f (z, ε) = f1 (z) + εf2 (z) + ε2 f3 (z) + · · · + εk−1 fk (z) + εk O(1),

1.3. Averaging theory for arbitrary order and dimension

57

where the function fi is the one deﬁned in (1.103) for i = 1, 2, . . . , k. From the assumption (ii) of the theorem we have that f (z, ε) = εr−1 fr (z) + · · · + εk−1 fk (z) + εk O(1). Clearly, x(·, z, ε) is a T -periodic solution if and only if f (z, ε) = 0, because x(t, z, ε) is deﬁned for all t ∈ [0, T ]. From the Brouwer degree theory (see Lemma 1.3.12 in Appendix 1.3.6) and hypothesis (ii) we have, for |ε| > 0 suﬃciently small, that dB (fr (z), V, a) = dB (f (z, ε), V, a) = 0. Hence, by Theorem 1.3.10(i) (see Appendix 1.3.6), 0 ∈ f (V, ε) for |ε| > 0 suﬃciently small, i.e, there exists aε ∈ V such that f (aε , ε) = 0. Therefore, for |ε| > 0 suﬃciently small, x(t, aε , ε) is a periodic solution of (1.100). Clearly, we can choose aε such that aε → a when ε → 0, because f (z, ε) = 0 in V \ {a}. This completes the proof of the theorem. For proving Theorem 1.3.6 we also need the following lemma. Lemma 1.3.9. Let w(·, z, ε) : [0, tˇz ] → Rn be the solution of the system w (t) =

k

εi [D2 ϕ(t, w)]−1 Fi (t, ϕ(t, w)) + εk+1 [D2 ϕ(t, w)]−1 R(t, ϕ(t, w), ε),

i=1

(1.125) such that w(0, z, ε) = z. Then, ψ(·, z, ε) : [0, t˜z ] → Rn deﬁned as ψ(t, z, ε) = ϕ (t, w(t, z, ε)) is the solution of (1.100) with ψ(0, z, ε) = z. Proof. Given z ∈ D, let M (t) = D2 ϕ(t, z). The result about diﬀerentiable dependence on initial conditions implies that the function M (t) is given as the fundamental matrix of the diﬀerential equation u = ∂F0 (t, ϕ(t, z))u. So the matrix M (t) is invertible for each t ∈ [0, T ]. Now the proof follows immediately from the derivative of ψ(t, ξ, ε) with respect to t. Proof of Theorem 1.3.6. Let x(·, z, ε) be a solution of (1.100) with x(0, z, ε) = z. For each z ∈ V , there exists ε1 > 0 such that if ε ∈ [−ε1 , ε1 ] then x(·, z, ε) is deﬁned in [0, T ]. Indeed, from Lemma 1.3.9, x(t, z, ε) = ϕ (t, w(t, z, ε)) for each z ∈ V , where w(·, z, ε) is the solution of (1.125). Moreover, for |ε1 | > 0 suﬃciently small, w(t, z, ε) ∈ W for each (t, z, ε) ∈ [0, T ] × V × [−ε1 , ε1 ]. Repeating the argument of the proof of Theorem 1.3.5, we can show that tˇz = T for every z ∈ V . Since ϕ(·, z) is deﬁned in [0,T] for every z ∈ W , it follows that t˜z = T , i.e., x(·, z, ε) is also deﬁned in [0, T ]. Now, denoting f (z, ε) = x(T, z, ε) − z, the proof follows like that of Theorem 1.3.5.

58

Chapter 1. The Averaging Theory for Computing Periodic Orbits

1.3.3 Computing formulae Now we shall illustrate how to compute the formulae from Theorems 1.3.5 and 1.3.6 for some k ∈ N. In Subsection 1.3.4 we compute the formulae when F0 = 0 for Theorem 1.3.5 up to k = 5. And in Subsection 1.3.5 we compute the formulae when F0 = 0 for Theorem 1.3.6 up to k = 4. First of all, from (1.104) we should determine the sets Sl for l = 1, 2, 3, 4, 5: S1 S2 S3 S4

= {1}, = {(0, 1), (2, 0)}, = {(0, 0, 1), (1, 1, 0), (3, 0, 0)}, = {(0, 0, 0, 1), (1, 0, 1, 0), (2, 1, 0, 0), (0, 2, 0, 0), (4, 0, 0, 0)}.

To compute Sl it is convenient to exhibit a table of possibilities with the value bi in the column i. We start from the last column. Clearly, the last column can be ﬁlled only by zeroes and ones because 5b5 > 5 for b5 > 1; the same happens with the fourth and the third column, because 3b3 , 4b4 > 5, for b3 , b4 > 1. Taking b5 = 1, the unique possibility is b1 = b2 = b3 = b4 = 0, thus any other solution satisﬁes b5 = 0. Taking b5 = 0 and b4 = 1, the unique possibility is b1 = 1 and b2 = b3 = 0, thus any other solution must have b4 = b5 = 0. Finally, taking b5 = b4 = 0 and b3 = 1, we have two possibilities either b1 = 2 and b2 = 0, or b1 = 0 and b2 = 1. Thus any other solution satisﬁes b3 = b4 = b5 = 0. Now we observe that the second column can be ﬁlled only by 0, 1 or 2, since 2b2 > 5 for b2 > 2; and taking b3 = b4 = b5 = 0 and b2 = 1 the unique possibility is b1 = 3. Taking b3 = b4 = b5 = 0 and b2 = 2 the unique possibility is b1 = 1, thus any other solution satisﬁes b2 = b3 = b4 = b5 = 0. Finally, taking b2 = b3 = b4 = b5 = 0 the unique possibility is b1 = 5. Therefore the complete table of solutions is b1 0 1 0 S5 = 2 3 1 5

b2 0 0 1 0 1 2 0

b3 0 0 1 1 0 0 0

b4 0 1 0 0 0 0 0

b5 1 0 0 0 0 0 0

Now we can use (1.104) and (1.103) to compute the expressions of the yi ’s and fi ’s in each case.

1.3. Averaging theory for arbitrary order and dimension

59

1.3.4 Fifth order averaging of Theorem 1.3.5 Let us assume that F0 ≡ 0. From (1.104) we obtain the functions yi (t, z) for k = 1, 2, 3, 4, 5: t y1 (t, z) = F1 (s, z)ds, 0 t ∂F1 (s, z)y1 (s, z) ds, y2 (t, z) = 2F2 (s, z) + 2 ∂x 0 t ∂F2 y3 (t, z) = (s, z)y1 (t, z) 6F3 (s, z) + 6 ∂x 0 ∂ 2 F1 ∂F1 2 (s, z)y (s, z) + 3 +3 (s, z) y (s, z) ds, 1 2 ∂x2 ∂x t ∂F3 (s, z)y1 (s, z) y4 (t, z) = 24F4 (s, z) + 24 ∂x 0 ∂ 2 F2 ∂F2 (s, z)y2 (s, z) + 12 (s, z)y1 (s, z)2 + 12 ∂x2 ∂x ∂ 2 F1 (s, z)y1 (s, z) y2 (s, z) + 12 ∂x2 ∂ 3 F1 ∂F1 3 +4 (s, z)y (s, z) + 4 (s, z) ds, (s, z)y 1 3 ∂x3 ∂x t ∂F4 (s, z)y1 (s, z y5 (t, z) = 120F5 (s, z) + 120 ∂x 0 ∂ 2 F3 + 60 (s, z)y1 (s, z)2 ∂x2 ∂F3 ∂ 2 F2 (s, z)y2 (s, z) + 60 + 60 (s, z)y1 (s, z) y2 (s, z) ∂x ∂x2 ∂F2 ∂ 3 F2 (s, z)y3 (s, z) (s, z)y1 (s, z)3 + 20 + 20 ∂x3 ∂x ∂ 2 F1 (s, z)y1 (s, z) y3 (s, z) + 20 ∂x2 ∂ 2 F1 ∂ 3 F1 (s, z)y2 (s, z)2 + 30 (s, z)y1 (s, z)2 y2 (s, z) + 15 2 ∂x ∂x3 ∂ 4 F1 ∂F1 4 +5 (s, z)y4 (s, z) ds. (s, z)y1 (s, z) + 5 ∂x4 ∂x Therefore, from (1.103) we have that f0 (z) =0, f1 (z) = 0

T

F1 (t, z)dt,

60

Chapter 1. The Averaging Theory for Computing Periodic Orbits

∂F1 F2 (t, z)ds + (t, z)y1 (t, z) dt, ∂x 0 T ∂F2 (t, z)y1 (t, z) f3 (z) = F3 (t, z) + ∂x 0 1 ∂ 2 F1 1 ∂F1 2 + (t, z)y (t, z)y (t, z) + (t, z) dt, 1 2 2 ∂x2 2 ∂x T ∂F3 F4 (t, z) + (t, z)y1 (t, z) f4 (z) = ∂x 0 1 ∂ 2 F2 1 ∂F2 (t, z)y1 (t, z)2 + + (t, z)y2 (t, z) 2 2 ∂x 2 ∂x 1 ∂ 2 F1 + (t, z)y1 (t, z) y2 (t, z)dt 2 ∂x2 1 ∂ 3 F1 1 ∂F1 3 + (t, z)y3 (t, z) dt, (t, z)y1 (t, z) + 6 ∂x3 6 ∂x T ∂F4 (t, z)y1 (t, z) f5 (z) = F5 (t, z) + ∂x 0 1 ∂ 2 F3 1 ∂F3 + (t, z)y2 (t, z) (t, z)y1 (t, z)2 + 2 2 ∂x 2 ∂x 1 ∂ 2 F2 + (t, z)y1 (t, z) y2 (t, z) 2 ∂x2 3 1 ∂ F2 1 ∂F2 + (t, z)y3 (t, z) (t, z)y1 (t, z)3 + 3 6 ∂x 6 ∂x 1 ∂ 2 F1 (t, z)y1 (t, z) y3 (t, z) + 6 ∂x2 1 ∂ 2 F1 1 ∂ 3 F1 2 + (t, z)y (t, z) + (t, z)y1 (t, z)2 y2 (t, z) 2 8 ∂x2 4 ∂x3 1 ∂ 4 F1 1 ∂F1 4 (t, z)y + (t, z)y (t, z) + (t, z) dt. 1 4 24 ∂x4 24 ∂x

T

f2 (z) =

1.3.5 Fourth order averaging of Theorem 1.3.6 Now we assume that F0 ≡ 0. First a Cauchy problem or, equivalently, an integral equation (see Remark 1.3.4) must be solved to compute the expressions yi (t, z) for i = 1, 2, . . . , k. We give the integral equations and its solutions for k = 1, 2, 3, 4. Let η(t, z) be the function deﬁned in (1.107), and let M (z) = η(T, z). Hence, from (1.104) and (1.103) we obtain the functions y1 (t, z) and f1 (z): t y1 (t, z) = 0

∂F0 (s, ϕ(s, z))y1 (s, z) ds, F1 (s, ϕ(s, z)) + ∂x

1.3. Averaging theory for arbitrary order and dimension so

t

y1 (t, z) = eη(t,z)

e−η(s,z) F1 (s, ϕ(s, z))ds,

and

f1 (z) = M (z)

T

e−η(t,z) F1 (t, ϕ(t, z))dt.

Similarly, the functions y2 (t, z) and f2 (z) are given by: t ∂F1 (s, ϕ(s, z))y1 (s, z) y2 (t, z) = 2F2 (s, ϕ(s, z)) + 2 ∂x 0

∂ 2 F0 ∂F0 2 (s, ϕ(s, z)) y2 (s, z) dt, + (s, ϕ(s, z))y1 (s, z) + ∂x2 ∂x

so

∂F1 (s, ϕ(s, z))y1 (s, z) y2 (t, z) =e e 2F2 (s, ϕ(s, z)) + 2 ∂x 0 ∂ 2 F0 (s, ϕ(s, z))y1 (s, z)2 ds, ∂x2

t

η(t,z)

−η(s,z)

and

∂F1 (t, ϕ(t, z))y1 (t, z) F2 (t, ϕ(t, z)) + f2 (z) =M (z) e ∂x 0 1 ∂ 2 F0 (t, ϕ(t, z))y1 (t, z)2 dt. 2 ∂x2

T

−η(t,z)

The functions y3 (t, z) and f3 (z) are given by t ∂F2 (s, ϕ(s, z))y1 (s, z) 6F3 (s, ϕ(s, z)) + 6 y3 (t, z) = ∂x 0 ∂ 2 F1 ∂F1 +3 (s, ϕ(s, z)) y2 (s, z) (s, ϕ(s, z))y1 (s, z)2 + 3 2 ∂x ∂x ∂ 2 F0 +3 (s, ϕ(s, z))y1 (s, z) y2 (s, z) ∂x2 ∂ 3 F0 ∂F0 3 (s, ϕ(s, z))y3 (s, z) ds, (s, ϕ(s, z))y1 (s, z) + + ∂x3 ∂x

61

62

Chapter 1. The Averaging Theory for Computing Periodic Orbits

so

∂F2 (s, ϕ(s, z))y1 (s, z) y3 (t, z) =e 6F3 (s, ϕ(s, z)) + 6 e ∂x 0 ∂F1 ∂ 2 F1 (s, ϕ(s, z)) y2 (s, z) (s, ϕ(s, z))y1 (s, z)2 + 3 +3 ∂x2 ∂x 2 ∂ F0 (s, ϕ(s, z))y1 (s, z) y2 (s, z) +3 ∂x2 ∂ 3 F0 3 (s, ϕ(s, z))y (s, z) ds, + 1 ∂x3

t

η(t,z)

−η(s,z)

and

∂F2 e F3 (t, ϕ(t, z)) + (t, ϕ(t, z))y1 (t, z) f3 (z) =M (z) ∂x 0 1 ∂ 2 F1 1 ∂F1 (t, ϕ(t, z)) y2 (t, z) (t, ϕ(t, z))y1 (t, z)2 + + 2 2 ∂x 2 ∂x 1 ∂ 2 F0 + (t, ϕ(t, z))y1 (t, z) y2 (t, z) 2 ∂x2 1 ∂ 3 F0 3 + (t, ϕ(t, z))y1 (t, z) ds. 6 ∂x3

T

−η(t,z)

Finally, the functions y4 (t, z) and f4 (z) are given by t y4 (t, z) =

so

∂F3 (s, ϕ(s, z))y1 (s, z) ∂x 0 ∂ 2 F2 ∂F2 (s, ϕ(s, z))y2 (s, z) + 12 (s, ϕ(s, z))y1 (s, z)2 + 12 ∂x2 ∂x ∂ 2 F1 + 12 (s, ϕ(s, z))y1 (s, z) y2 (s, z) ∂x2 ∂ 3 F1 ∂F1 (s, ϕ(s, z))y1 (s, z)3 + 4 (s, ϕ(s, z))y3 (s, z) +4 ∂x3 ∂x ∂ 2 F0 +4 (s, ϕ(s, z))y1 (s, z) y3 (s, z) ∂x2 ∂ 2 F0 ∂ 3 F0 +3 (s, ϕ(s, z))y2 (s, z)2 ds + 6 (s, ϕ(s, z))y1 (s, z)2 y2 (s, z) 2 ∂x ∂x3 ∂ 4 F0 ∂F0 4 (s, ϕ(s, z))y4 (s, z) ds, + (s, ϕ(s, z))y1 (s, z) + ∂x4 ∂x 24F4 (s, ϕ(s, z)) + 24

1.3. Averaging theory for arbitrary order and dimension

63

∂F3 y4 (t, z) =e (s, ϕ(s, z))y1 (s, z) e 24F4 (s, ϕ(s, z)) + 24 ∂x 0 ∂ 2 F2 ∂F2 (s, ϕ(s, z))y2 (s, z) + 12 (s, ϕ(s, z))y1 (s, z)2 + 12 2 ∂x ∂x ∂ 2 F1 (s, ϕ(s, z))y1 (s, z) y2 (s, z) + 12 ∂x2 ∂ 3 F1 ∂F1 (s, ϕ(s, z))y3 (s, z) +4 (s, ϕ(s, z))y1 (s, z)3 + 4 3 ∂x ∂x ∂ 2 F0 +4 (s, ϕ(s, z))y1 (s, z) y3 (s, z) ∂x2 ∂ 2 F0 ∂ 3 F0 2 +3 (s, ϕ(s, z))y (s, z) ds + 6 (s, ϕ(s, z))y1 (s, z)2 y2 (s, z) 2 3 ∂x2 ∂x ∂ 4 F0 4 (s, ϕ(s, z))y (s, z) ds, + 1 ∂x4

t

−η(s,z)

η(t,z)

and

T

∂F3 (t, ϕ(t, z))y1 (t, z) ∂x 0 1 ∂ 2 F2 1 ∂F2 + (t, ϕ(t, z))y2 (t, z) (t, ϕ(t, z))y1 (t, z)2 + 2 2 ∂x 2 ∂x 1 ∂ 2 F1 (t, ϕ(t, z))y1 (t, z) y2 (t, z) + 2 ∂x2 1 ∂ 3 F1 1 ∂F1 (t, ϕ(t, z))y3 (t, z) + (t, ϕ(t, z))y1 (t, z)3 + 6 ∂x3 6 ∂x 2 1 ∂ F0 (t, ϕ(t, z))y1 (t, z) y3 (t, z) + 6 ∂x2 1 ∂ 2 F0 1 ∂ 3 F0 + (t, ϕ(t, z))y2 (t, z)2 ds + (t, ϕ(t, z))y1 (t, z)2 y2 (t, z) 2 3 8 ∂x 4 ∂x 1 ∂ 4 F0 4 + (t, ϕ(t, z))y1 (t, z) ds. 24 ∂x4

f4 (z) =M (z)

e

−η(t,z)

F4 (t, ϕ(t, z)) +

1.3.6 Appendix: basic results on the Brouwer degree In this appendix we present the existence and uniqueness result from the degree theory in ﬁnite dimensional spaces. We follow Browder’s paper [11], where the properties of the classical Brouwer degree are formalized. We also present some results we shall need for proving our main results. Theorem 1.3.10. Let X = Rn = Y for a given positive integer n. For bounded open subsets V of X, consider continuous mappings f : V → Y , and points y0 in Y such that y0 does not lie in f (∂V ) (as usual, ∂V denotes the boundary of V ).

64

Chapter 1. The Averaging Theory for Computing Periodic Orbits

Then to each such triple (f, V, y0 ), there corresponds an integer d(f, V, y0 ) having the following three properties: (i) If d(f, V, y0 ) = 0 then y0 ∈ f (V ). If f0 is the identity map Y then, of X onto for every bounded open set V and y0 ∈ V , we have d f0 V , V, y0 = ±1. (ii) (Additivity) If f : V → Y is a continuous map with V a bounded open set in X, and V1 and V2 are a pair of disjoint open subsets of V such that / f (V \(V1 ∪ V2 )) then, d (f0 , V, y0 ) = d (f0 , V1 , y0 ) + d (f0 , V1 , y0 ). y0 ∈ (iii) (Invariance under homotopy) Let V be a bounded open set in X, and consider a continuous homotopy {ft : 0 ≤ t ≤ 1} of maps of V into Y . Let {yt : 0 ≤ / ft (∂V ) for any t ∈ [0, 1]. t ≤ 1} be a continuous curve in Y such that yt ∈ Then d(ft , V, yt ) is constant in t on [0, 1]. Theorem 1.3.11. The degree function d(f, V, y0 ) is uniquely determined by the conditions of Theorem 1.3.10. For the proofs of Theorems 1.3.10 and 1.3.11, see [11]. Lemma 1.3.12. We consider continuous functions fi : V → Rn , for i = 0, 1, . . . , k, and f, g, r : V × [ε0 , ε0 ] → Rn , given by g(·, ε) = f1 (·) + εf2 (·) + ε2 f3 (·) + · · · + εk−1 fk (·), f (·, ε) = g(·, ε) + εk r(·, ε). Assume that g(z, ε) = 0 for all z ∈ ∂V and ε ∈ [−ε0 , ε0 ]. If for |ε| > 0 suﬃciently small dB (f (·, ε), V, y0 ) is well deﬁned, then dB (f (·, ε), V, y0 ) = dB (g(·, ε), V, y0 ). For a proof of Lemma 1.3.12, see [14, Lemma 2.1].

1.4 Three applications of Theorem 1.3.5 The ﬁrst application studies the periodic solutions of the H´enon–Heiles Hamiltonian using the averaging theory of second order. The other two examples analyze the limit cycles of some classes of polynomial diﬀerential systems in the plane. These last two applications use the averaging theory of third order. More precisely, these three applications are based in Theorem 1.3.5. In the next subsection we summarize the results of Theorem 1.3.5 up to third order, precisely the ones used in the applications here considered.

1.4.1 The averaging theory of ﬁrst, second and third order As far as we know, the averaging theory of third order for studying speciﬁcally periodic orbits was developed by ﬁrst time in [14]. Now we summarize it here from Theorem 1.3.5 which is given at any order.

1.4. Three applications of Theorem 1.3.5

65

Consider the diﬀerential system x(t) ˙ = εF1 (t, x) + ε2 F2 (t, x) + ε3 F3 (t, x) + ε4 R(t, x, ε),

(1.126)

where F1 , F2 , F3 : R × D → R and R : R × D × (−εf , εf ) → R are continuous functions, T -periodic in the ﬁrst variable, and D is an open subset of Rn . Assume that the following hypotheses (i) and (ii) hold: (i) F1 (t, ·) ∈ C 2 (D), F2 (t, ·) ∈ C 1 (D) for all t ∈ R, F1 , F2 , F3 , R, Dx2 F1 ,Dx F2 are locally Lipschitz with respect to x, and R is twice diﬀerentiable with respect to ε. We deﬁne Fk0 : D → R for k = 1, 2, 3 as 1 T F10 (z) = F1 (s, z)ds, T 0 1 T F20 (z) = [Dz F1 (s, z) · y1 (s, z) + F2 (s, z)] ds, T 0 1 T 1 ∂ 2 F1 1 ∂F1 y1 (s, z)T (s, z)y2 (s, z) F30 (z) = (s, z)y1 (s, z) + 2 T 0 2 ∂z 2 ∂z ∂F2 (s, z)(y1 (s, z)) + F3 (s, z) ds, + ∂z where

s

y1 (s, z) =

F1 (t, z)dt, 0

y2 (s, z) =

s

∂F1 (t, z) ∂z

t

F1 (r, z)dr + F2 (t, z) dt. 0

(ii) For V ⊂ D an open and bounded set, and for each ε ∈ (−εf , εf ) \ {0} there exists aε ∈ V such that F10 (aε ) + εF20 (aε ) + ε2 F30 (aε ) = 0 and dB (F10 + εF20 + ε2 F30 , V, aε ) = 0. Then for |ε| > 0 suﬃciently small there exists a T -periodic solution ϕ(·, ε) of the system such that ϕ(0, ε) = aε . The expression dB (F10 + εF20 + ε2 F30 , V, aε ) = 0 means that the Brouwer degree of the function F10 + εF20 + ε2 F30 : V → Rn at the ﬁxed point aε is not zero. A suﬃcient condition for the inequality to be true is that the Jacobian of the function F10 + εF20 + ε2 F30 at aε is not zero. If F10 is not identically zero, then the zeros of F10 + εF20 + ε2 F30 are mainly the zeros of F10 for ε suﬃciently small. In this case, the previous result provides the averaging theory of ﬁrst order. If F10 is identically zero and F20 is not identically zero, then the zeros of F10 + εF20 + ε2 F30 are mainly the zeros of F20 for ε suﬃciently small. In this case, the previous result provides the averaging theory of second order.

66

Chapter 1. The Averaging Theory for Computing Periodic Orbits

If F10 and F20 are both identically zero and F30 is not identically zero, then the zeros of F10 + εF20 + ε2 F30 are mainly the zeros of F30 for ε suﬃciently small. In this case, the previous result provides the averaging theory of third order.

1.4.2 The H´enon–Heiles Hamiltonian The results presented in this subsection have been proved by Jim´enez–Llibre [50]. The classical H´enon–Heiles potential consists of a two dimensional harmonic potential plus two cubic terms. It was introduced in 1964, as a model for studying the existence of a third integral of motion of a star in a rotating meridian plane of a galaxy in the neighborhood of a circular orbit [40]. The classical H´enon–Heiles potential has been generalized by introducing two parameters to each cubic term, 1 2 1 (p + p2y + x2 + y 2 ) + Bxy 2 + Ax3 , 2 x 3

(1.127)

such that B = 0, with x, y, px , py ∈ R. Then the classical H´enon–Heiles Hamiltonian system corresponds to A = −1, B = 1. It is given by x˙ = px , p˙ x = −x − (Ax2 + By 2 ), y˙ = py ,

(1.128)

p˙ y = −y − 2Bxy. As usual, the dot denotes derivative with respect to the independent variable t ∈ R, the time. We name (1.128) the H´enon–Heiles Hamiltonian systems with two parameters, or simply the H´enon–Heiles systems. The periodic orbits in the H´enon–Heiles potential have been numerically studied and classiﬁed by Churchill–Pecelli–Rod [20], Davies–Huston–Baranger [24] and others [10, 31, 74]. Maciejewski–Radzki–Rybicki [68] did an analytical study of a more general H´enon–Heiles Hamiltonians including a third cubic term of the form Cx2 y, which can be removed by a proper rotation, and two more parameters associated with the quadratic part of the potential. They proved the existence of connected branches of non-stationary periodic orbits in the neighborhood of a given degenerate stationary point. Theorem 1.4.1. At every positive energy level the H´enon–Heiles Hamiltonian system (1.128) has at least (i) one periodic orbit if (2B − 5A)(2B − A) < 0 (see Figure 1.1), (ii) two periodic orbits if A + B = 0 and A = 0 (this case contains the classical H´enon–Heiles system), and (iii) three periodic orbits if B(2B − 5A) > 0 and A + B = 0 (see Figure 1.2).

1.4. Three applications of Theorem 1.3.5

67

%

$

Figure 1.1: Open region (2B − 5A)(2B − A) < 0 in the parameter space (A, B), where there is at least one periodic orbit with multipliers diﬀerent from 1.

%

$

Figure 1.2: Open region B(2B − 5A) > 0 and A + B = 0 in the parameter space (A, B), where there are at least three periodic orbits with multipliers diﬀerent from 1. When A + B = 0, there are at least two periodic orbits with multipliers diﬀerent from 1.

Proof. For proving this theorem we shall apply Theorem 1.3.5 to the Hamiltonian system (1.128). Generically, the periodic orbits of a Hamiltonian system with more than one degree of freedom are on cylinders fulﬁlled by periodic orbits. Therefore we cannot apply directly Theorem 1.3.5 to a Hamiltonian system, since the Jacobian of the function f at the ﬁxed point a will always be zero. Then we must apply Theorem 1.3.5 to every Hamiltonian ﬁxed level, where the periodic orbits generically are isolated. On the other hand, in order to apply Theorem 1.3.5 we need a small parameter ε. So in the Hamiltonian system (1.128) we change the variables (x, y, px , py ) to (X, Y, pX , pY ) where x = εX, y = εY , px = εpX and py = εpY . In the new

68

Chapter 1. The Averaging Theory for Computing Periodic Orbits

variables, the system (1.128) becomes X˙ = pX , p˙ X = −X − ε(AX 2 + BY 2 ), Y˙ = pY , p˙Y = −Y − 2εBXY.

(1.129)

This system again is Hamiltonian with Hamiltonian

1 1 2 (pX + p2Y + X 2 + Y 2 ) + ε BXY 2 + AX 3 . 2 3

(1.130)

As the change of variables is only a scale transformation, for all ε diﬀerent from zero, the original and the transformed systems (1.128) and (1.129) have essentially the same phase portrait and, additionally, the system (1.129) for ε suﬃciently small is close to an integrable one. First we change the Hamiltonian (1.130) and the equations of motion (1.129) to polar coordinates for ε = 0, which is an harmonic oscillator. Thus we have X = r cos θ,

pX = r sin θ,

Y = ρ cos(θ + α),

pY = ρ sin(θ + α).

Recall that this is a change of variables when r > 0 and ρ > 0. Moreover, doing this change of variables, the angular variables θ and α appear in the system. Later on, the variable θ will be used for obtaining the periodicity necessary for applying the averaging theory. The ﬁxed value of the energy in polar coordinates is

1 3 1 Ar cos3 θ + Brρ2 cos θ cos2 (θ + α) , (1.131) h = (r2 + ρ2 ) + ε 2 3 and the equations of motion are given by r˙ = −ε sin θ A r2 cos2 θ + B ρ2 cos2 (θ + α) ,

2 ˙θ = −1 − ε cos θ A r cos2 θ + ρ B cos2 (θ + α) , r ρ˙ = −εB rρ cos θ sin(2(θ + α)), cos θ 2 A r cos2 θ + B(ρ2 − 2r2 ) cos2 (θ + α) . α˙ = ε r

(1.132)

However, the derivatives of the left hand side of these equations are with respect to the time variable t, which is not periodic. We change to the θ variable as the independent one, and we denote by a prime the derivative with respect to θ. The angular variable α cannot be used as the independent variable since the new

1.4. Three applications of Theorem 1.3.5

69

diﬀerential system would not have the form (1.98) for applying Theorem 1.3.5. The system (1.132) goes over to ε r sin θ A r2 cos2 θ + Bρ2 cos2 (θ + α) , r = r + ε(A r2 cos3 θ + Bρ2 cos θ cos2 (θ + α)) ε Br2 ρ cos θ sin(2(θ + α)) ρ = , r + ε(A r2 cos3 θ + Bρ2 cos θ cos2 (θ + α)) ε cos θ B ρ2 − 2r2 cos2 (θ + α) + A r2 cos2 θ . A = − r + ε(Bρ2 cos θ cos2 (θ + α) + A r2 cos3 θ)

Of course this system has now only three equations because we do not need the θ equation. If we write the previous system as a Taylor series in powers of ε, we have r = ε sin θ(A r2 cos2 θ + Bρ2 cos2 (θ + α)) 2 sin 2θ 2 A r (1 + cos(2θ)) + Bρ2 (1 + cos(2(θ + α)) + O(ε3 ), − ε2 8r ρ = εBrρ cos θ sin(2(θ + α)) − ε2 Bρ cos2 θ sin(2(θ + α))(Ar2 cos2 θ + Bρ2 cos2 (2(θ + α))) + O(ε3 ), (1.133) cos θ (A r2 cos2 θ + B(ρ2 − 2r2 ) cos2 (θ + α)) A = −ε r cos2 θ + ε2 (Ar2 cos2 θ + Bρ2 cos2 (θ + α)) r2 · (A r2 cos2 θ + B(ρ2 − 2r2 ) cos2 (θ + α)) + O(ε3 ). Now system (1.133) is 2π-periodic in the variable θ. In order to apply Theorem 1.3.5 we must ﬁx the value of the ﬁrst integral at h > 0 and, by solving equation (1.131) for ρ, we obtain ρ=

h − r2 /2 − εA r3 cos3 θ/3 . 1/2 + ε B r cos θ cos2 (θ + α)

(1.134)

Then, substituting ρ in equations (1.133), we obtain the two diﬀerential equations r = ε sin θ(A r2 cos2 θ + B(2h − r2 ) cos2 (θ + α)) sin 2θ 2 − ε2 A r2 (1 + cos(2θ)) + B 2h − r2 (1 + cos(2(θ + α))) 8r (1.135) 2 + AB r3 sin θ cos3 θ cos2 (θ + α) 3 + 2B 2 hr sin(2θ) cos4 (θ + α) − B 2 r3 sin(2θ) cos4 (θ + α)

+ O(ε3 ),

70

Chapter 1. The Averaging Theory for Computing Periodic Orbits

and

B 2 2 3 α =ε (3r − 2h) cos θ cos (θ + α) − Ar cos θ r 2 + ε2 (A2 r2 cos6 θ + AB(6h − 5r2 ) cos4 θ cos2 (θ + α) 3 B2 + 2 (r2 − 2h)2 cos2 θ cos4 (θ + α) ) + O(ε3 ). r

(1.136)

Clearly, equations (1.135) and (1.136) satisfy the assumptions of Theorem 1.3.5, and it has the form of (1.98) with F1 = (F11 , F12 ) and F2 = (F21 , F22 ), where F11 = sin θ A r2 cos2 θ + B(2h − r2 ) cos2 (θ + α) , B F12 = (3r2 − 2h) cos θ cos2 (θ + α) − Ar cos3 θ, r and 2 sin 2θ 2 A r (1 + cos(2θ)) + B 2h − r2 (1 + cos(2(θ + α))) 8r 2 − AB r3 sin θ cos3 θ cos2 (θ + α) − 2B 2 hr sin(2θ) cos4 (θ + α) 3 + B 2 r3 sin(2θ) cos4 (θ + α), 2 = A2 r2 cos6 θ + AB(6h − 5r2 ) cos4 θ cos2 (θ + α) 3 B2 + 2 (r2 − 2h)2 cos2 θ cos4 (θ + α). r

F21 = −

F22

As r = 0 the functions F1 and F2 are analytical. Furthermore, they are 2π-periodic in the variable θ, the independent variable of the system (1.135)-(1.136). However, the averaging theory of ﬁrst order does not apply because the average functions of F1 and F2 in the period vanish, 2π (F11 , F12 ) dθ = (0, 0) . f1 (r, A) = 0

As the function f1 from Theorem 1.3.5 is zero, we procede to calculate the function f2 by applying the second order averaging theory. We have that f2 is deﬁned by f2 (r, A) =

2π

[DrA F1 (θ, r, A).y1 (θ, r, A) + F2 (θ, r, A)] dθ,

where

y1 (θ, r, A) =

θ

F1 (t, r, A) dt . 0

(1.137)

1.4. Three applications of Theorem 1.3.5

71

The two components of the vector y1 are θ F11 (t, r, A) dt y11 = 0

1 B(2h−r2 ) sin2 (θ/2) cos(2(θ + α))+2 cos(2α+θ)+3 −Ar2 (cos3 θ − 1) , = 3 and

θ

F12 (t, r, A) dt

y12 = 0

Ar Bh (9 sin θ + sin 3θ)− (3 sin(2α + θ)+sin(2α + 3θ) − 4 sin 2α + 6 sin θ) 12 6r Br + (3 sin(2α + θ) + sin(2α + 3θ) − 4 sin(2α) + 6 sin θ). 4

=−

For the Jacobian matrix

⎛

∂F11 ⎜ ∂r DrA F1 (θ, r, A) = ⎜ ⎝ ∂F12 ∂r

⎞ ∂F11 ∂A ⎟ ⎟, ∂F12 ⎠ ∂A

we obtain ⎛ ⎞ 2Ar cos2 θ − 2Br cos2 (θ + α) sin θ −2B(2h − r2 ) cos(θ + α) sin θ sin(θ + α) ⎜ ⎟ ⎜ −A cos3 θ + 6B cos2 (θ + α) cos θ − 2B (3r2 −2h) cos θ cos(θ + α) sin(θ + α)⎟ ⎜ ⎟. r ⎝ ⎠ 2 B 2 − 2 3r −2h cos (θ + α) cos θ r We can now calculate the function (1.137) from Theorem 1.3.5, and we obtain Br f2 = − (6B − A)(r2 − 2h) sin 2A, 12 1 2 r (5A2 − 12AB − 3B 2 ) − 2B(A − 6B)(h − r2 ) cos(2α) + 2Bh(6A − B) . 12 We have to ﬁnd the zeros (r∗ , A∗ ) of f2 (r, A), and to check that the Jacobian determinant (1.138) |Dr,A f2 (r∗ , A∗ )| = 0. Solving the equation f2 (r, A) = 0, we obtain ﬁve solutions (r∗ , A∗ ) with r∗ > 0, namely

√ 2Bh 14Bh B(A − 6B) ,0 , , ±π/2 . , 2h, ± arcsec 4B 2 + 6AB − 5A2 3B − A 9B − 5A (1.139)

72

Chapter 1. The Averaging Theory for Computing Periodic Orbits

The ﬁrst two solutions are not good, because for them we would get from (1.134) that ρ = 0 when ε = 0, and ρ must be positive. The third solution exists if B(3B − A) > 0. The last two solutions exist if B(9B − 5A) > 0. The Jacobian (1.138) of the third solution is −

5B 2 h2 (A − 6B)(A − 2B)(A + B) 9(A − 3B)

(1.140)

and, for the last two solutions, the Jacobian coincides and is equal to 7B 2 h2 (A − 6B)(5A − 2B)(A − B) . 9(5A − 9B)

(1.141)

Summarizing, from Theorem 1.3.5 the third solution of f2 (r, A) = 0 provides a periodic orbit for the system (1.135)-(1.136) (and consequently of the Hamiltonian system (1.129) on the Hamiltonian level h > 0) if B(3B−A) > 0, (A−6B)(A− 2B)(A + B) = 0, and from (1.134) we get ρ = 2(A − 2B)h/(A − 3B); we also need (2B−A)(3B−A) > 0. The conditions B(3B−A) > 0 and (2B−A)(3B−A) > 0 can be reduced to B(2B − A) > 0, where (A − 6B)(A − 2B) = 0 is included, but A + B = 0 is not. Then the third solution provides a periodic orbit when B(2B − A) > 0 and A + B = 0. In a similar way the last two solutions of f2 (r, A) = 0 provide two periodic orbits for the system (1.135)-(1.136) if B(9B − 5A) > 0, (A − 6B)(5A − 2B)(A − B) = 0, and from (1.134) we get ρ = 2(5A − 2B)h/(5A − 9B); we also need (2B−5A)(9B−5A) > 0. The conditions B(9B−5A) > 0 and (2B−5A)(9B−5A) > 0 can be reduced to B(2B − 5A) > 0, where the condition (A − 6B)(5A − 2B)(A − B) = 0 is included. Then the fourth and ﬁfth solutions provide two periodic orbits whenever B(2B − 5A) > 0. There is one periodic orbit if the third solution exists, and the last two solutions do not. There are two periodic orbits if the two last solutions exist, and not the third one, i.e., when A + B = 0. Finally, there are three periodic orbits if the third, fourth and ﬁfth solutions exist. Now the statements of Theorem 1.3.5 follow easily. The regions in the parameter space where periodic orbits exist are summarized in Figures 1.1 and 1.2.

1.4.3 Limit cycles of polynomial diﬀerential systems The results presented in this subsection come from Llibre–Swirszcz [57]. After the deﬁnition of limit cycle due to Poincar´e [75], the statement of the 16-th Hilbert’s problem [41], and the discovery by Li´enard [52] that limit cycles are important in nature, the study of limit cycles of planar diﬀerential systems has been one of the main problems of the qualitative theory of diﬀerential equations. One of the best ways of producing limit cycles is by perturbing the periodic orbits of a center. This has been studied intensively perturbing the periodic orbits

1.4. Three applications of Theorem 1.3.5

73

of the centers of the quadratic polynomial diﬀerential systems, see the book of Christopher–Li [18] and the references quoted there. It is well known that if a quadratic polynomial diﬀerential system has a limit cycle, this must surround a focus. Up to now, the maximum number of known limit cycles surrounding a focus of a quadratic polynomial diﬀerential system is 3, which coincides with the maximum number of small limit cycles which can bifurcate by Hopf from a singular point of a quadratic polynomial diﬀerential system, see Bautin [4]. But, as far as we know, up to now there are few quadratic centers for which it is proved that the perturbation of their periodic orbits inside the class of all quadratic polynomial diﬀerential systems can produce 3 limit cycles. These are the center whose exterior boundary is formed by three invariant straight lines (see ˙ la¸ dek [88]), three diﬀerent families of reversible quadratic centers (see Swirszcz ´ Zo [83]), and the center x˙ = −y(1+x), y˙ = x(1+x) (see Buic˘ a–Gasull–Yang [13]). The study of the perturbation of this last center has been made through the Melnikov function of third order, computed using the algorithm developed by Fran¸coise [35] and Iliev [44]. Here, we can provide a new and shorter proof of this second result by using the averaging theory, see Theorem 1.4.2. We study the limit cycles of the following two diﬀerential systems: the quadratic systems ¯ + Cy ¯ 2 ), ¯ 2 + Bxy x˙ = −y(1 + x) + ε(λx + Ax ¯ 2 + Exy ¯ + F¯ y 2 ), y˙ = x(1 + x) + ε(λy + Dx

(1.142)

such that, for ε = 0, have a straight line consisting of singular points, and the cubic systems of the form x˙ = −y(1 − x2 − y 2 ) + ε3 λx +

3

εs

s=1

y˙ =

x(1 − x − y ) + ε λy + 2

2

3

3 s=1

ε

s

3 i=0 3

ai,s xi y 3−i , (1.143) i 3−i

bi,s x y

,

i=0

such that, for ε = 0, have a unit circle consisting of singular points. Note that the perturbation of these cubic systems is inside the class of all polynomial diﬀerential system with linear and cubic homogeneous non-linearities. We study for ε = 0 suﬃciently small the number of limit cycles of the systems (1.142) and (1.143) bifurcating from the periodic orbits of the centres of (1.142) and (1.143) for ε = 0, respectively. Our main results are the following. ¯ B, ¯ C, ¯ D, ¯ E¯ and F¯ , the system (1.142) has Theorem 1.4.2. For convenient λ, A, 3 limit cycles bifurcating from the periodic orbits of the center for ε = 0. Theorem 1.4.3. The following statements hold for system (1.143): (i) using the averaging theory of third order for ε = 0 suﬃciently small, we can obtain at most 5 limit cycles of the system (1.143) bifurcating from the periodic orbits of the center located at the origin of system (1.143) with ε = 0;

74

Chapter 1. The Averaging Theory for Computing Periodic Orbits

(ii) for convenient λ, ai,s , bi,s , i = 0, 1, 2, 3, s = 1, 2, 3, the system (1.143) has 0, 1, 2, 3, 4 or 5 limit cycles bifurcating from the periodic orbits of the center for ε = 0. It is known that systems of the form x˙ = −y + P3 (x, y), y˙ = x + Q3 (x, y), with P3 and Q3 homogeneous polynomials of degree 3 can have 5 small limit cycles bifurcating by Hopf from the origin, see [62, 81]. We are going to use the following result due to Cherkas [17]. Lemma 1.4.4. The diﬀerential equation λr + a(ϕ)rk dr = dϕ 1 + b(ϕ)rk−1 after the change of variable ρ(ϕ) =

r(ϕ)k−1 1 + b(ϕ)r(ϕ)k−1

becomes the Abel equation dρ = (k − 1)b(ϕ)(λb(ϕ) − a(ϕ))ρ3 dϕ + [(k − 1)(a(ϕ) − 2λb(ϕ)) − b (ϕ)] ρ2 + (k − 1)λρ. Combining Lemma 1.4.4 with polar coordinates transformation we immediately get the next result. Corollary 1.4.5. Let P (x, y) and Q(x, y) be homogenous polynomials of degree n. Then the diﬀerential system x˙ = −y + λx + Pn (x, y) y˙ = x + λy + Qn (x, y) can be transformed into the Abel equation dρ = (k − 1)B(ϕ)(λB(ϕ) − A(ϕ))ρ3 dϕ + [(k − 1)(A(ϕ) − 2λB(ϕ)) − B (ϕ)] ρ2 + (k − 1)λρ, where A(ϕ) = cos ϕPn (cos ϕ, sin ϕ) + sin ϕQn (sin ϕ, cos ϕ), and B(ϕ) = cos ϕQn (cos ϕ, sin ϕ) − sin ϕPn (sin ϕ, cos ϕ).

(1.144)

1.4. Three applications of Theorem 1.3.5

75

Proof. The system (1.144) expressed in polar coordinates becomes r˙ = λr + A(ϕ)rn , y˙ = 1 + B(ϕ)rn . Dividing r˙ by ϕ˙ and using Lemma 1.4.4 the proof of the corollary follows.

Proof of Theorem 1.4.2. From Corollary 1.4.5 applied to the system (1.142), it follows that ﬁnding limit cycles of (1.142) is equivalent to ﬁnding periodic solutions of 1 dρ ¯ − 4λ) cos ϕ = (sin ϕ)ρ2 + − cos ϕ((3A¯ + C¯ + E dϕ 4 ¯ cos 3ϕ + (A¯ − C¯ − E) ¯ +D ¯ + F¯ + (B ¯ +D ¯ − F¯ ) cos 2ϕ) sin ϕ)ρ3 + 2(B ¯ cos 3ϕ + ((A¯ + C¯ − 2λ) cos ϕ + (A¯ − C¯ − E)

(1.145)

¯ + F¯ ) sin ϕ + (B ¯ +D ¯ − F¯ ) sin 3ϕ)ρ2 + λρ . + (D We are going to apply Theorem 1.3.5 to system (1.145). We ﬁrst solve the diﬀerential equation dρ = (sin ϕ)ρ2 , dϕ with initial condition ρ(0) = R/(1 + R), and we get ρ(ϕ, R) = R/(1 + R cos ϕ). Thus MR (ϕ) in (1.37) will be a solution of the diﬀerential equation MR (ϕ) = (2R sin ϕ)/(1 + R cos ϕ), namely, MR (ϕ) = 1 + 2 ln(1 + R) − 2 ln(1 + r cos ϕ). Thus, formula (1.37) yields 2π R F (R) = λ Ξ(ϕ, R) 0 cos ϕ(R cos ϕ + 8 cos(2ϕ) + 3R cos(3ϕ))R2 + A¯ 4Ξ(ϕ, R) 2 ¯ (2R sin 2ϕ + 8 sin 3ϕ + 3R sin 4ϕ)R +B 8Ξ(ϕ, R) 2

2

cos ϕ(3R cos ϕ + 4) sin ϕR − C¯ Ξ(ϕ, R) 2 ϕ(3R cos ϕ + 4) sin ϕR2 cos ¯ +D Ξ(ϕ, R) cos ϕ(R cos ϕ + 8 cos 2ϕ + 3R cos 3ϕ − 4)R2 ¯ −E 4Ξ(ϕ, R) (5R cos ϕ + 8 cos 2ϕ + 3R cos 3ϕ) sin ϕR2 ¯ +F dϕ, 4Ξ(ϕ, R)

(1.146)

76

Chapter 1. The Averaging Theory for Computing Periodic Orbits

where Ξ(ϕ, R) = (R cos ϕ + 1)3 (2 log(R + 1) − 2 log(R cos ϕ + 1) + 1). Now, observe ¯ D ¯ and F¯ are odd π-periodic functions of ϕ, thus their that the terms in front of B, integrals from 0 to 2π are equal to zero. Therefore,

F (R) =

2π

λ

R Ξ(ϕ, R) 0 cos ϕ(R cos ϕ + 8 cos(2ϕ) + 3R cos(3ϕ))R2 + A¯ 4Ξ(ϕ, R)

cos ϕ(3R cos ϕ + 4) sin2 ϕR2 Ξ(ϕ, R) 2 ¯ cos ϕ(R cos ϕ + 8 cos 2ϕ + 3R cos 3ϕ − 4)R dϕ +E 4Ξ(ϕ, R) ¯ ¯ ¯ 4 (R). = λf1 (R) + Af2 (R) + Cf3 (R) − Ef + C¯

(1.147)

We claim that the four functions f1 , f2 , f3 and f4 are linearly independent. Now we prove the claim. By straightforward calculation we obtain the following Taylor expansions: 1 πR 2615R4 − 800R3 + 312R2 − 96R + 48 + O(R6 ), 24 1 πR3 313R2 − 60, R − 18 + O(R6 ), f2 (R) = 24 1 πR3 401R2 − 84R − 6 + O(R6 ), f3 (R) = 24 1 f4 (R) = − πR3 43R2 − 12R + 6 + O(R6 ). 24 f1 (R) =

The determinant of the coeﬃcient matrix of terms R2 , . . . , R5 is π 4 /3 and the claim follows. A well-known classical result states that if a family of n functions is linearly independent, then there exists a linear combination of them with at least n − 1 zeroes. Thus, Theorem 1.4.2 follows. Proof of Theorem 1.4.3. First we prove statement (ii). We shall use third order averaging to show that the system x˙ = −y(1 − x2 − y 2 ) + ε3 λx 1 (75Bε + 108E + 19840)εx3 + (j + 24)εx2 y − 1200

27B (81E + 16480)ε 3 2 −C + + 4ε (A − 4λ) + ε xy 2 128 300 1 + ε(2j + Dε)y 3 , 2

(1.148)

1.4. Three applications of Theorem 1.3.5

77

x(1 − x2 − y 2 ) + ε3 λy

3B (81E + 18080)ε 1 3 2 + x2 y + (Dε − 2j)εx + ε C − (1.149) 2 128 300 1 (27E + 6560)εy 3, − (j + 40)εxy 2 − 300 can have 0, 1, 2, 3, 4 or 5 limit cycles for an appropriate choice of the parameters λ, A, B, C, D and E. The system (1.148)-(1.149) is, clearly, a special case of (1.143); thus this will prove statement (ii). Using Cherkas Transformation (see Lemma 1.4.4) we transform the system (1.148)-(1.149) into the Abel equation y˙ =

dρ = εF1 + ε2 F2 + ε3 F3 , dϕ where

(1.150)

3 16 (3E + 640) cos(4ϕ) + 8(sin(2ϕ) − 2 sin(4ϕ)) − cos(2ϕ) 50 3

16 9 cos(2ϕ) , + ρ2 − (3E + 640) cos(4ϕ) − 8 sin(2ϕ) + 48 sin(4ϕ) + 50 3 3 ρ F2 = 25(6400j + 75B + 432E + 117760) cos(2ϕ) 30000

F1 = ρ3

− 75 cos(4ϕ)(72(j + 8)E + 15360(j + 8) − 25B) − 600 sin(2ϕ)(400j + 25D + 12E + 7360) + 480000(j + 8) sin(4ϕ) − 7200(E + 80) sin(6ϕ) + 3(9E + 1120)(9E + 2720) sin(8ϕ) − 400(27E + 7360) cos(6ϕ) + 14400(3E + 640) cos(8ϕ)

3 3B + ρ2 − C cos(2ϕ) − B cos(4ϕ) + 3D sin(ϕ) cos(ϕ) , 128 16 F3 = −2λρ

+ ρ2 (A − 4λ)(2 cos(2ϕ) − 3 cos(4ϕ)) + A # 4D 11B + 2C − + 2λ + ρ3 A cos 4ϕ − A − 64 3 1 + sin(2ϕ)(384(100(j + 4)D − 3C(3E + 640)) + B(513E + 103040)) 76800 − 96 cos(2ϕ)(25(2j − 7)B + 3200C − 6D(3E + 640)) − 400 cos(4ϕ)(3(4j + 21)B + 128(3C + 2D + 6λ)) + sin(6ϕ)(1152(3CE + 640C − 400D) − B(81E + 23680))

78

Chapter 1. The Averaging Theory for Computing Periodic Orbits − 96 cos(6ϕ)(175B − 640(5C + 18D) − 54DE) + 800 sin(4ϕ)(11B + 64(3D − 2C)) + 144B(3E + 640) sin(8ϕ) $ + 38400B cos(8ϕ) .

By straightforward calculations, we verify that F10 = 0, ρ3 sin ϕ((27E + 4160) cos ϕ + 3(3(3E + 640) cos 3ϕ − 800 sin 3ϕ)) 300 ρ2 − 2 sin(2ϕ)(27(3E + 640) cos 2ϕ−800(9 sin 2ϕ + 1))+4800 sin2 ϕ , 600 and F20 = 0. Next, 1 2 ρ (9B cos ϕ + 12B cos(3ϕ) + 128C cos ϕ − 192D sin ϕ) sin ϕ y2 (ρ, ϕ) = 128 8j B 9E 128 3 +ρ + − + sin(2ϕ) 3 32 25 15 1 (400j + 25D − 24E + 1280) sin2 ϕ − 50 8 9 48 jE sin(4ϕ) + (9j + 494) sin2 (2ϕ) − j sin(4ϕ) − 200 9 5 1 81E 2 sin2 (4ϕ) 4 216 B sin(4ϕ) + − E sin2 (3ϕ) + E sin2 (4ϕ) + 64 4000 5 25 63 3 9 − E sin(4ϕ) − E sin(6ϕ) + E sin(8ϕ) − 64 sin2 (3ϕ) 25 5 5 1472 7904 3808 2 sin (4ϕ) − sin(4ϕ) − sin(6ϕ) + 384 sin(8ϕ) + 5 15 9 1 29 243E 2 sin2 (4ϕ) + ρ4 − − (21E + 2480) sin2 ϕ + E sin2 (3ϕ) 16000 25 25 27 1 162 E sin2 (4ϕ) + (189E + 9920) sin(2ϕ) + E sin(4ϕ) − 25 300 25 87 27 1528 2 464 2 + E sin(6ϕ) − E sin(8ϕ) − sin (2ϕ) + sin (3ϕ) 100 20 9 5 2856 2 3056 10672 − sin (4ϕ) + sin(4ϕ) + sin(6ϕ) − 288 sin(8ϕ) 5 15 45 y1 (ρ, ϕ) =

+ ρ5 and

((27E + 4160) cos ϕ+3(3(3E + 640) cos(3ϕ)−800 sin(3ϕ)))2 sin2 ϕ 60000

2D F30 (ρ) = −2λρ + Aρ − A − B − − 2λ ρ3 3

9E 91B 7D 4E 4 −C+ − ρ + D− ρ5 + Eρ6 . − 128 3 5 5 2

1.4. Three applications of Theorem 1.3.5

79

The coeﬃcients of F30 are linearly independent (linear) functions of λ, A, B, C, D and E. Therefore, for any ρ1 , ρ2 , ρ3 , ρ4 , ρ5 ∈ R there exist λ, A, B, C, D, E such that F30 (ρi ) = 0 for i = 1, 2, 3, 4, 5. This ends the proof of (ii). Now we sketch the proof of statement (i). If, instead of doing the computations of the proof of statement (ii) for the system (1.148)-(1.149), we did them for the general system (1.143) we would obtain a function F30 (ρ) which again is a polynomial of degree 6 in ρ without independent term. Thus the averaging theory of third order can only produce for ε = 0 suﬃciently small at most 5 limit cycles for system (1.143), bifurcating from the periodic orbits at the origin of system (1.143) with ε = 0.

1.4.4 The generalized polynomial diﬀerential Li´enard equation The results in this subsection have been proved by Llibre–Mereu–Teixeira [54]. The second part of the Hilbert’s problem is related with the least upper bound on the number of limit cycles of polynomial vector ﬁelds having a ﬁxed degree. The generalized polynomial Li´enard diﬀerential equations x¨ + f (x)x˙ + g(x) = 0,

(1.151)

was introduced in [52]. Here, the dot denotes diﬀerentiation with respect to the time t, and f (x) and g(x) are polynomials in the variable x of degrees n and m, respectively. For this subclass of polynomial vector ﬁelds we have a simpliﬁed version of Hilbert’s problem, see [53, 80]. In 1977 Lins–de Melo–Pugh [53] studied the classical polynomial Li´enard diﬀerential equations (1.151) obtained when g(x) = x, and stated the following conjecture: “if f (x) has degree n ≥ 1 and g(x) = x, then (1.151) has at most [n/2] limit cycles”. They also proved the conjecture for n = 1, 2. The conjecture for n ∈ {3, 4, 5} is still open. For n ≥ 5 this conjecture is not true as it has been proved recently by Dumortier–Panazzolo–Roussarie [29], and De Maesschalck–Dumortier [25]. Recently, this conjecture has been proved for n = 3, see Chengzhi–Llibre [61]. So, at this moment, it only remains to know if the conjecture holds or not for n = 4. We note that a classical polynomial Li´enard diﬀerential equation has a unique singular point. However, it is possible for generalized polynomial Li´enard diﬀerential equations to have more than one singular point. Many of the results on the limit cycles of polynomial diﬀerential systems have been obtained by considering limit cycles which bifurcate from a single degenerate singular point; these are the so called small amplitud limit cycles, see ˆ [60]. We denote by H(m, n) the maximum number of small amplitude limit cyˆ cles for systems of the form (1.151). The values of H(m, n) give a lower bound for the maximum number H(m, n) (i.e., the Hilbert number ) of limit cycles that the diﬀerential equation (1.151) with m and n ﬁxed can have. The ﬁniteness of

80

Chapter 1. The Averaging Theory for Computing Periodic Orbits

H(m, n) for every positive integers m and n is unknown. For more information about Hilbert’s 16-th problem and related topics, see [46, 50]. n 1

2

3

4

5

6 7

8

9

10 11 12 13 . . . 48 49 50

1

1*

1

2

2

3 3

4

4

5

5

2

1* 1*

2

3

3

4 5

5

6

7

3

1*

2

2

4

4

6 6

6

8

8

4

2

3

4

4

6

7 8

9

9

10 11 12 13

5

2

3

4

6

6

8 9 10 11

6

3

4

6

7

8

8 9 9 9

7

3

5

6

8

9

m 8

4

5

6

9

10

9

4

6

8

9

11

10

5

7

8

10

11

5

7

8

11

12

6

8

10 12

13 .. .

6 .. .

9 .. .

10 13 .. .

6

6

· · · 24 24 →

7

8

9

· · · 32 33 →

8

10 10 · · · 36 38 38

20 10 13 14 17 .. .. .. .. . . . . 48 24 32 36 49 24 33 38 50

↓

↓

38

ˆ Table 1.1: The values of H(m, n) or H(m, n) for Li´enard systems. Now we shall describe, brieﬂy, the main results about the limit cicles on Li´enard diﬀerential systems: x (i) in 1928 Li´enard [52] proved that, if m = 1 and F (x) = 0 f (s)ds is a continuous odd function having a unique root at x = a and being monotone increasing for x ≥ a, then equation (1.151) has a unique limit cycle; x (ii) in 1973 Rychkov [77] proved that, if m = 1 and F (x) = 0 f (s)ds is an odd polynomial of degree ﬁve, then equation (1.151) has at most two limit cycles; (iii) in 1977 Lins–de Melo–Pugh [53] proved that H(1, 1) = 0 and H(1, 2) = 1; (iv) in 1998 Coppel [22] proved that H(2, 1) = 1; (v) Dumortier, Li and Rousseau in [27] and [30] proved that H(3, 1) = 1:

1.4. Three applications of Theorem 1.3.5

81

(vi) in 1997 Dumortier–Li [28] proved that H(2, 2) = 1. Up to now and as far as we know, the four cases (iii) to (vi) (marked with asterisks in Table 1.1) are the only ones for which Hilbert numbers H(m, n) are determined. Blows, Lloyd and Lynch, in the papers [6], [61] and [64] have used inductive arguments in order to prove the following results: ˆ (vii) if g is odd then H(m, n) = [n/2]; ˆ (viii) if f is even then H(m, n) = n, whatever g is; ˆ (ix) if f is odd then H(m, 2n + 1) = [(m − 2)/2] + n; ˆ (x) if g(x) = x + ge (x), where ge is even, then H(2m, 2) = m. Christopher–Lynch [19, 67] and Lynch [65, 66] have developed a new algebraic method for determining the Liapunov quantities of system (1.151) and proved the following: ˆ (xi) H(m, 2) = [(2m + 1)/3]; ˆ n) = [(2n + 1)/3]; (xii) H(2, ˆ (xiii) H(m, 3) = 2[(3m + 2)/8] for all 1 < m ≤ 50; ˆ (xiv) H(3, n) = 2[(3n + 2)/8] for all 1 < m ≤ 50; ˆ k) = H(k, ˆ ˆ 6) = (xv) the values in Table 1.1 for H(4, 4), k = 6, 7, 8, 9 and H(5, ˆ 5). H(6, ˆ 6), H(6, ˆ 7), In 1998 Gasull–Torregrosa [36] obtained upper bounds for H(7, ˆ ˆ H(7, 7) and H(4, 20). ˆ ˆ In 2006 the values in Table 1.1 for H(m, n) = H(n, m), for n = 4, m = 10, 11, 12, 13; n = 5, m = 6, 7, 8, 9; and n = 6, m = 5, 6 were given by Yu–Han in [87]. By using the averaging theory, we shall study the maximum number of limit ˜ cycles H(m, n) which can bifurcate from the periodic orbits of a linear center perturbed inside the class of all generalized polynomial Li´enard diﬀerential equations of degrees m and n as follows: x˙ = y, k y˙ = −x − εk (fnk (x)y + gm (x)),

(1.152)

k≥1 k where for every k the polynomials gm (x) and fnk (x) have degree m and n respectively, and ε is a small parameter; i.e., the maximal number of medium amplitude limit cycles which can bifurcate from the periodic orbits of the linear center x˙ = y, y˙ = −x, perturbed as in (1.152). ˜ In fact, we shall mainly compute lower estimations of H(m, n). More pre˜ cisely, we compute the maximum number of limit cycles Hk (m, n) which bifurcate

82

Chapter 1. The Averaging Theory for Computing Periodic Orbits

from the periodic orbits of the linear center x˙ = y, y˙ = −x, using the averaging ˜ ˜ k (m, n) ≤ H(m, n) ≤ H(m, n). Note theory of order k, for k = 1, 2, 3. Of course, H that, up to now, there were no known lowers estimations for H(m, n) when (i) m = 4 and n > 13, or m > 20 and n = 4, (ii) m = 5 and n > 9, or m > 9 and n = 5, (iii) m = 6 and n > 7, or m > 7 and n = 6, (iv) m, n > 7. After our results we will have lowers estimations of H(m, n) for all m, n ≥ 1. ˆ ˜ k (m, n) ≤ H(m, n) for k = 1, 2, 3 for the From these estimations we obtain that H ˆ values which H(m, n) is known. k Theorem 1.4.6. If for every k = 1, 2, 3, the polynomials fnk (x) and gm (x) have degree n and m respectively, with m, n ≥ 1, then for |ε| suﬃciently small, the maximum number of medium limit cycles of the polynomial Li´enard diﬀerential systems (1.152) bifurcating from the periodic orbits of the linear center x˙ = y, y˙ = −x, using the averaging theory ˜ 1 (m, n) = n ; (i) of ﬁrst order is H 2 # m n $ n−1 ˜ + , ; and (ii) of second order is H2 (m, n) = max 2 2 2 ˜ 3 (m, n) = n + m − 1 . (iii) of third order is H 2

From Theorem 1.4.6, Table 1.2 follows immediately. ˆ It seems that the numbers H(m, n) can be symmetric with respect to m and n. Some studies is this direction are made in [63]. We remark that in general ˜ k (n, m) for k = 1, 2, but H ˜ 3 (m, n) = H ˜ 3 (n, m). ˜ k (m, n) = H H Proof of Theorem 1.4.6(i). We shall need the ﬁrst order averaging theory. In order to apply the ﬁrst order averaging method we write (1.152) with k = 1 in polar coordinates, (r, θ), where x = r cos θ, y = r sin θ, r > 0. In this way, (1.152) is written in the standardform for applying the averaging theory. If we write f (x) = n m i i i=0 ai x and g(x) = i=0 bi x , then the system (1.152) becomes r˙ = −ε

n i=0

ai r

i+1

i

2

cos θ sin θ +

m i=0

i

i

bi r cos θ sin θ ,

(1.153)

1.4. Three applications of Theorem 1.3.5

83

n 1

2

3

4

5

6

7

8

9

10 11 12 13 . . . 48 49 50

1

1

1

2

2

3

3

4

4

5

5

6

6

· · · 24 24 →

2

1

1

2

2

3

3

4

4

5

5

6

6

7

· · · 24 25 →

3

1

2

2

3

3

4

4

5

5

6

6

7

7

· · · 25 25 →

4

2

2

3

3

4

4

5

5

6

6

7

7

8

· · · 25 26 →

5

2

3

3

4

4

5

5

6

6

7

7

8

8

· · · 26 26 →

6

3

3

4

4

5

5

6

6

7

7

8

8

9

· · · 26 27 →

7

3

4

4

5

5

6

6

7

7

8

8

9

9

· · · 27 27 →

m 8

4

4

5

5

6

6

7

7

8

8

9

9

10 · · · 27 28 →

9

4

5

5

6

6

7

7

8

8

9

9

10 10 · · · 28 28 →

10

5

5

6

6

7

7

8

8

9

9

10 10 11 · · · 28 29 →

11

5

6

6

7

7

8

8

9

9

10 10 11 11 · · · 29 29 →

12

6

6

7

7

8

8

9

9 10 10 11 11 12 · · · 29 30 →

13 .. .

6 .. .

7 .. .

7 .. .

8 .. .

8 .. .

9 .. .

9 .. .

10 10 11 11 12 12 · · · 30 30 → .. .. .. .. .. .. .. .. .. .. . . . . . . . . . .

20 10 10 11 11 12 12 13 13 14 14 15 15 16 · · · 33 34 → .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 48 24 24 25 25 26 26 27 27 28 28 29 29 30 · · · 47 48 → 49 24 25 25 26 26 27 27 28 28 29 29 30 30 · · · 48 48 → 50

↓

↓

↓

↓

↓

↓

↓

↓

↓

↓

↓

↓

↓

↓

↓

↓

˜ 3 (m, n). The numbers written in roman style (like “6”) Table 1.2: Values of H coincide with the ones of Table 1.1. The numbers written in italic (like “6 ”) are smaller than the corresponding of Table 1.1. Finally, the numbers written in boldface (like “6”) are unknown in Table 1.1. ε θ˙ = −1 − r

n

ai r

i+1

cos

i+1

θ sin θ +

m

i

i+1

bi r cos

θ .

(1.154)

i=0

i=0

Taking θ as the new independent variable, system (1.153)-(1.154) becomes n m dr =ε ai ri+1 cosi θ sin2 θ + bi ri cosi θ sin θ + O(ε2 ), dθ i=0 i=0 and 1 F10 (r) = 2π

2π

n i=0

ai r

i+1

i

2

cos θ sin θ +

m i=0

i

i

bi r cos θ sin θ dθ.

84

Chapter 1. The Averaging Theory for Computing Periodic Orbits In order to calculate the exact expression of F10 we use the following formulas

2π

cos2k+1 θ sin2 θdθ = 0,

k = 0, 1, . . .

cos2k θ sin2 θdθ = α2k = 0,

k = 0, 1, . . .

2π

2π

cosk θ sin θdθ = 0,

k = 0, 1, . . .

Hence, F10 (r) =

n 1 ai αi ri+1 . 2 i=0

(1.155)

i even

Then the polynomial F10 (r) has at most [n/2] positive roots, and we can choose the coeﬃcients ai with i even in such a way that F10 (r) has exactly [n/2] simple positive roots. Hence, statement (i) of Theorem 1.4.6 is proved.

Proof of Theorem 1.4.6(ii). We shall now use the second order averaging theory. i If we f1 (x) = ni=0 ai xi , f2 (x) = ni=0 ci xi , g1 (x) = m i=0 bi x and write m i g2 (x) = i=0 di x , then system (1.152) with k = 2 in polar coordinates (r, θ), r > 0 becomes r˙ = − ε

n

ai ri+1 cosi θ sin2 θ +

i=0

−ε

2

ε θ˙ = − 1 − r

bi ri cosi θ sin θ

i=0 n

ci r

i=0 n

i=0 n 2

ε − r

m

i+1

2

cos θ sin θ +

m

i

i

di r cos θ sin θ ,

i=0

ai r

ci r

i=0

i

i+1

i+1

i+1

cos

i+1

cos

θ sin θ +

θ sin θ +

m

i=0 m

i

i+1

bi r cos

θ

i

i+1

di r cos

θ .

i=0

Taking θ as the new independent variable system, (1.156) is written dr = εF1 (θ, r) + ε2 F2 (θ, r) + O(ε3 ), dθ

(1.156)

1.4. Three applications of Theorem 1.3.5

85

where F1 (θ, r) =

n

ai ri+1 cosi θ sin2 θ +

i=0 n

bi ri cosi θ sin θ,

i=0 m

F2 (θ, r) =

m

ci ri+1 cosi θ sin2 θ +

i=0

− r sin θ cos θ

di ri cosi θ sin θ

i=0 n

ai ri cosi θ sin θ +

i=0

m

2 bi ri−1 cosi θ

.

i=0

Now we determine the corresponding function F20 . For this we compute n m d F1 (θ, r) = (i + 1)ai ri cosi θ sin2 θ + ibi ri−1 cosi θ sin θ, dr i=1 i=0

θ

F1 (φ, r)dφ which is equal to

and 0

a1 r2 (α11 sin θ + α21 sin(3θ)) + · · · + al rl+1 α1l sin θ + α2l sin(3θ) + · · · + α( l+3 )l sin((l + 2)θ 2

+ a0 r (α10 θ + α20 sin(2θ)) + · · · + ab rb+1 α1b θ + α2b sin(2θ) + · · · + α( b+4 )b sin(b + 2)θ 2

1 m m+1 (1 − cos θ) , b0 (1 − cos θ) + · · · + bm r m+1

(1.157)

where l is the greatest odd number less than or equal to n, b is the greatest even number less than or equal to n, and αij are real constants exhibited during the θ computation of 0 cosi φ sin2 φ dφ for all i. We know from (1.155) that F10 is identically zero if and only if ai = 0 for all i even. Moreover, 0

2π

cosi θ sin3 θdθ = 0,

i = 0, 1, . . .

cosi θ sin2 θ sin((2k + 1)θ)dθ = 0,

i, k = 0, 1, . . .

0 2π

2π

cos2i+1 θ sin2 θdθ = 0, 2π

cos2i θ sin2 θdθ = A2i = 0,

i = 0, 1, . . . i = 0, 1, . . .

86

Chapter 1. The Averaging Theory for Computing Periodic Orbits

0 2π

2π

cosi θ sin θdθ = 0,

i = 0, 1, . . .

0 2π

2k+1 cos2i θ sin θ sin((2k + 1)θ)dθ = B2i = 0,

cos2i+1 θ sin θ sin((2k + 1)θ)dθ = 0,

i, k = 0, 1, . . . i, k = 0, 1, . . .

So 2π 0

=

d F1 (θ, r)y1 (θ, r)dθ dr

k

l

−

j=2 i=1 j even i odd

+

k

i+1 ai bj ri+j j+1

l

cosi+j+1 θ sin2 θdθ

2π

jai bj ri+j

2π

cosj θ sin θ α1i sin θ + · · · + α i+3 i sin((i + 2)θ) dθ 2

j=2 i=1 j even i odd

=r α ˜ 10 a1 b0 + (˜ α12 a1 b2 + α ˜ 30 a3 b0 )r2 + · · · +

α ˜ij ai bj rl+k−1 ,

i+j=l+k

1+i Ai+j+1 +j α1i Bj1 + α2i Bj2 + · · · + α i+3 i Bji+2 , for all i, j and 2 j+i k being the greatest even number less than or equal to m. Moreover, where α ˜ ij = −

2π

F2 (θ, r)dθ = 0

b

k

cosi θ sin2 θdθ

i=0 i even

+

2π

ci ri+1 l

j=0 i=1 j even i odd

2ri+j ai bj

2π

cosi+j+1 θ sin2 θdθ

= A0 c0 r + · · · + Ab cb rb+1 + 2 A2 a1 b0 r+A4 (a3 b0 + a1 b2 )r3 +· · ·+Al+k+1 rl+k ai b j . i+j=l+k

Then F20 (r) is the polynomial r ρ10 a1 b0 + (ρ12 a1 b2 + ρ30 a3 b0 )r2 + (ρ14 a1 b4 + ρ32 a3 b2 + ρ50 a5 b0 )r4 + · · · + ρlk al bk rl+k−1 + A0 c0 + A2 c2 r2 + · · · + Ab cb rb ,

(1.158)

where ρij = α ˜ ij + 2Ai+j+1 for all i, j. Note that, in order to ﬁnd the positive roots of F20 , we must ﬁnd the zeros of a polynomial in r2 of degree equal to

1.4. Three applications of Theorem 1.3.5

87

max {(l + k − 1)/2, b/2}. We have that b/2 = [n/2] and (l + k − 1)/2 = [(n − 1)/2] + [m/2]; see Table 1.3: n

m

odd even

l

k

(l + k − 1)/2

[(n − 1)/2] + [m/2]

n

m

(n + m − 1)/2

(n − 1)/2 + m/2

m

(n − 1 + m − 1)/2

((n − 1) − 1)/2 + m/2

m-1

(n + m − 1 − 1)/2

(n − 1)/2 + (m − 1)/2

even even n-1 odd odd

n

even odd n-1 m-1 (n − 1 + m − 1 − 1)/2 ((n − 1) − 1)/2 + (m − 1)/2 Table 1.3: Values of (l + k − 1)/2 written using the integer part function. We conclude that F20 has at most max{[(n − 1)/2] + [m/2], [n/2]} positive roots. Moreover, we can choose the coeﬃcients ai , bj , ck in such a way that (1.158) has exactly max{[(n − 1)/2] + [m/2], [n/2]} simple positive roots. Hence, the statement (ii) of Theorem 1.4.6 follows. Proof of Theorem 1.4.6(iii). We shall now use the third order averagingtheory. n n n i i i If we write f (x) = a x , f (x) = 1 i 2 i=0 i=0 i=0 pi x , m cimx , f3i(x) = m i i g1 (x) = i=0 bi x , g2 (x) = i=0 di x and g3 (x) = i=0 qi x , then an equivalent system to (1.152) with k = 3 will be found by considering polar coordinates (r, θ). So, r˙ = − sin θ εA + ε2 B + ε3 C , (1.159) cos θ εA + ε2 B + ε3 C , θ˙ = − 1 − r where A= B=

n i=0 n

ai r

i+1

i

cos θ sin θ +

ci ri+1 cosi θ sin θ +

i=0

C=

n i=0

pi ri+1 cosi θ sin θ +

m i=0 m i=0 m

bi ri cosi θ, di ri cosi θ, qi ri cosi θ.

i=0

Taking θ as the new independent variable system, (1.159) becomes

A2 cos θ sin θ dr 2 =εA sin θ + ε B sin θ − dθ r (1.160)

3 2 cos θ sin θ A 2AB cos θ sin θ 3 + C sin θ . +ε − r2 r

88

Chapter 1. The Averaging Theory for Computing Periodic Orbits

By (1.155), we know that F10 is identically zero if and only if ai = 0 for all i even; and by (1.158) we obtain that F20 is identically zero if and only if the coeﬃcients ai , bj and ck satisfy cμ =

1 Aμ

ρi,j ai bj ,

(1.161)

i+j=μ+1 i odd, j even

where μ is even, and Aμ and ρi,j are given in Subsection 1.3.2. In order to apply the third order averaging method we need to compute the corresponding function F30 . So, the proof of statement (iii) of Theorem 1.4.6 will be a direct consequence of the next auxiliary lemmas. The proof of the next lemma is straightforward and follows from some tedious computations; it will be omitted. Lemma 1.4.7. The corresponding functions y1 (θ, r) and y2 (θ, r) of the third order averaging method are expressed by (1.157) and y2 (θ, r) = C0 + C1 r + C2 r2 + · · · + Cλ rλ , respectively, where λ = max{2n + 1, 2m − 1} and C2k+1 = c0ij ai aj + d0ij bi bj + i+j+=2k

+

i+j=2k+2

fij0 ai aj θ2

i+j=2k+1

ai b j

k+1

ai b j +

i+j+=2k+1

C2k =

+

+

cos(2i + 1)θ

i=0

k+1

ai bj θ + d2k+1

a02i+2

cos(2i + 2)θ

i=0

i+j=2k+1

a12i+1 sin(2i + 1)θ

i+j+=2k−1

bi bj +

i+j=2k+1

bi bj

k+1 i=0

a12i+2 sin(2i + 2)θ) ,

b02i+2

d1ij bi bj +

i+j=2k+1

ai aj +

i+j=2k−1

i+j+=2k+1

k+1 i=0

c1ij ai aj +

ai aj θ + c2k

i+j=2k

a02i+1

i=0

+

bi bj +

i+j=2k+2

i+j=2k+1

ai aj +

i+j+=2k

+

bi bj

k+1

i+j=2k+2

+

+ d2k+1 + c2k θ +

i+j=2k

e0ij ai bj θ

e1ij ai bj θ

i+j=2k

ai b j θ

i+j=2k

cos(2i + 2)θ

k+1 i=0

b02i+1

cos(2i + 1)θ

1.4. Three applications of Theorem 1.3.5 +

+

ai bj + c2k−1 +

i+j=2k

ai b j

k+1

al2i+2 ,

where k = 0, 1, . . . , λ/2.

ai b j θ

k+1

b12i+1

sin(2i + 1)θ

i=0

i+j=2k

b12i+2 sin(2i + 2)θ) ,

i=0

i+j+=2k

al2i+1 ,

89

bl2i+1 ,

al2i+2 ,

Lemma 1.4.8. The integral

2π 0

clij , dlij , elij , fijl are real constants for l = 1, 2 and

1 ∂ 2 F1 2 2 ∂r 2 (s, r)(y1 (s, r)) ds

equals the polynomial

π(D0 + D1 r + D2 r2 + · · · + Dκ rκ ), where

⎧ n + 2m − 1 ⎪ ⎪ ⎪ ⎨n + 2m − 2 κ= ⎪ 3n + 1 ⎪ ⎪ ⎩ 3n

and Dχ =

if if if if

m>n+1 m>n+1 m≤n+1 m≤n+1

1 βijk ai aj ak +

i+j+k=χ−1

and and and and

either m or n is even, both m and n are odd, n is even, n is odd,

for χ = 0, 1, . . . , κ, where

1 γijk ,

1 δijk

1 γijk ai b j b k +

i+j+k=χ+1 1 , βijk

(1.162)

1 δijk ai aj b k ,

i+j+k=χ

are real constants.

Proof. Let us write ∂ 2 F1 (s, r) = h1 (r) + h2 (r), ∂r2 where h1 (r) =

n

i(i + 1)ai ri−1 cosi θ sin2 θ,

i=1

h2 (r) =

m

i(i − 2)bi ri−2 cosi θ sin θ,

i=2

and (y1 (s, r))2 = g12 (r) + 2g1 (r)g2 (r) + g22 (r), with g1 (r) = s1 (r) + s2 (r), where s1 (r) =a1 r2 (α11 sin θ + α21 sin(3θ)) + · · · + al rl+1 α1l sin θ + α2l sin(3θ) + · · · + α( l+3 )l sin((l + 2)θ , 2

s2 (r) =a0 r (α10 θ + α20 sin(2θ)) + · · · + ab rb+1 α1b θ + α2b sin(2θ) + · · · + α( b+4 )b sin(b + 2)θ , 2

90

Chapter 1. The Averaging Theory for Computing Periodic Orbits

and

g2 (r) = b0 (1 − cos θ) + · · · + bm rm

1 (1 − cosm+1 θ) . m+1

Then, ∂ 2 F1 2 (s, r) (y1 (s, r)) =h1 (r) g12 (r) + 2g1 (r)g2 (r) + g22 (r) ∂r2 + h2 (r) g12 (r) + 2g1 (r)g2 (r) + g22 (r) . From

2π

cos2i θ sin2 θ sin(ρ1 θ) sin(ρ2 θ)dθ = M1 (2i, ρ1 , ρ2 ) = 0,

ρ1 , ρ2 odd,

0 2π

cos2i+1 θ sin2 θ sin(ρ1 θ) sin(ρ2 θ)dθ = 0,

ρ1 , ρ2 odd,

for i = 1, 2, . . ., we have that

2π

h1 (r)s1 (r)2 dθ =

l l

b

1 ζijk ai aj ak ri−1 rj+1 rk+1 ,

i=2 k=1 j=1 k odd j odd i even

where 1 ζijk =

k+2

j+2

δρjk1 ρ2 i(i + 1)α ρ1 +1 j α ρ2 +1 k M1 (i, ρ1 , ρ2 ), 2

ρ1 =1 ρ =1 ρ odd ρ1 odd

and

) δρjk1 ρ2

=

2

1 if ρ1 = ρ2 and j = k, k. 2 if ρ1 = ρ2 or j =

2π Thus, H1 (r) = 0 h1 (r)s1 (r)2 dθ is a polynomial in r of degree 3n − 1 if n even, and 3n if n odd. Knowing that 0

0 2π

2π

cosi θ sin2 θ sin(ρ1 θ)θdθ = M2 (i, ρ1 , 0) = 0,

ρ1 odd,

0 2π

cos2i θ sin2 θ sin(ρ1 θ) sin(ρ2 θ)dθ = 0,

ρ1 odd, ρ2 even,

cos2i+1 θ sin2 θ sin(ρ1 θ) sin(ρ2 θ)dθ = M3 (2i, ρ1 , ρ2 ) = 0, ρ1 odd, ρ2 even,

1.4. Three applications of Theorem 1.3.5 for i = 1, 2, . . ., we have that 2π b 2h1 (r)s1 (r)s2 (r)dθ = 0

91

n l

2 ζijk ai aj ak ri−1 rj+1 rk+1

j=1 i=1 k=0 k even j odd b

+

l l

3 ζijk ai aj ak ri−1 rj+1 rk+1 ,

j=1 i=1 k=0 k even j odd i odd

where k+2

λ ζijk =

j+2

2i(i + 1)α ρ1 +1 j α ρ2 +2 k Mλ (i, ρ1 , ρ2 ), λ = 2, 3. 2

ρ1 =1 ρ2 =0 ρ1 odd ρ2 even

2

2π Thus, the degree of the polynomial H2 (r) = 0 2h1 (r)s1 (r)s2 (r)dθ in r is 3n. From 2π cosi θ(sin2 θ)θ2 dθ = M4 (i, 0, 0) = 0,

0 2π

0 2π

cos θ sin θ sin(ρ1 θ) sin(ρ2 θ)dθ = M5 (2i, ρ1 , ρ2 ) = 0, 2

2i

cos2i+1 θ sin2 θ sin(ρ1 θ) sin(ρ2 θ)dθ = 0,

2π

ρ1 , ρ2 even, ρ1 , ρ2 even,

cosi θ sin2 θ sin(ρ1 θ)θdθ = M6 (i, ρ1 , 0) = 0,

ρ1 even,

for i = 1, 2, . . ., we have that 2π b h1 (r)s22 (r)dθ = 0

n b

4 ζijk ai aj ak ri−1 rj+1 rk+1

j=0 i=1 k=0 k even j even

+

b

b

n

5 ζijk ai aj ak ri−1 rj+1 rk+1

j=1 i=2 k=0 k even j even i even

+

b

b n

6 ζijk ai aj ak ri−1 rj+1 rk+1 ,

j=0 i=1 k=0 k even j even

where λ ζijk =

k+2

j+2

ρ1 =0 ρ2 =0 ρ1 even ρ2 even

δρjk1 ρ2 i(i + 1)α ρ1 +2 j α ρ2 +2 k Mλ (i, ρ1 , ρ2 ), 2

2

λ = 4, 5, 6,

92

Chapter 1. The Averaging Theory for Computing Periodic Orbits

2π with δρjk1 ρ2 as above. Thus, H3 (r) = 0 h1 (r)s22 (r)dθ is a polynomial in r of degree 3n + 1 if n even, and 3n − 1 if n odd. Knowing that 2π cosi θ sin2 θ sin(ρ1 θ)dθ = 0, ρ1 = 1, 2, . . . 0

2π

cos2i θ(sin2 θ)θdθ = M7 (i, 0, 0) = 0,

2π

cos2i+1 θ(sin2 θ)θdθ = 0,

for i = 1, 2, . . ., we have that

n b m

2π

h1 (r)(s1 (r) + s2 (r))g2 (r)dθ = 0

k=0

7 ζijk ai aj bk ri−1 rj+1 rk ,

j=0 i=1 j even

2π 7 where k + i is odd, and ζijk = i(i + 1)α1j M7 (i, 0, 0). Thus, H4 (r) = 0 h1 (r)(s1 (r) + s2 (r))g2 (r)dθ is a polynomial in r of degree 2n + m − 1 if m is even, 2n + m if n is even and m is odd, and 2n + m − 2 if both n and m are odd. The equalities 2π cos2i θ sin2 θdθ = M8 (i, 0, 0) = 0, 0 2π

cos2i+1 θ sin2 θdθ = 0,

for i = 1, 2, . . . imply 0

2π

h1 (r)g22 (r)dθ =

m m n

8 ζijk ai bj bk ri−1 rj rk ,

k=0 j=0 i=1

8 where ζijk = δjk i(i + 1)M8 (i, 0, 0) with

) δjk =

1 if j = k, 2 if j = k.

2π Thus, H5 (r) = 0 h1 (r)g22 (r)dθ is a polynomial in r of degree 2m + n − 1 if n or m is even, and 2m + n − 2 if n and m are both odd. From 2π cosi θ sin θ sin(ρ1 θ) sin(ρ2 θ)dθ = 0, ρ1 , ρ2 odd 0

for i = 1, 2, . . ., we have that H6 (r) =

2π 0

h2 (r)s21 (r)dθ = 0.

1.4. Three applications of Theorem 1.3.5

93

From the values of the integrals 0 2π

0 2π

2π

cos2i θ(sin θ)θ sin(ρ1 θ)dθ = M9 (i, ρ1 , 0) = 0,

ρ1 odd,

cos2i+1 θ(sin θ)θ sin(ρ1 θ)dθ = 0,

ρ1 odd,

cosi θ sin θ sin(ρ1 θ) sin(ρ2 θ)dθ = 0,

ρ1 even, ρ2 odd,

for i = 1, 2, . . ., we have that

2π

h2 (r)s1 (r)s2 (r)dθ = 0

l

b

m

9 ζijk bi aj ak ri−2 rj+1 rk+1 ,

j=0 i=2 k=1 k odd j even i even

where 9 ζijk

=

l+2

i(i − 1)α1j α ρ1 +1 k M9 (i, ρ1 , 0). 2

ρ1 =1 ρ1 odd

2π Thus, H7 (r) = 0 h2 (r)s1 (r)s2 (r)dθ is a polynomial in r of degree 2n + m − 1 if m even, and 2m + n − 2 if m odd. The formulas

0 2π

0 2π

2π

cosi θ(sin θ)θ2 dθ = M10 (i, 0, 0) = 0,

0 2π

cos2i θ(sin θ)θ sin(ρ1 θ)dθ = 0,

ρ1 even,

cos2i+1 θ(sin θ)θ sin(ρ1 θ)dθ = M11 (i, ρ1 , 0) = 0,

cosi θ sin θ sin(ρ1 θ) sin(ρ2 θ)dθ = 0,

ρ1 even, ρ1 , ρ2 odd,

for i = 1, 2, . . . imply 0

2π

h2 (r)s22 (r)dθ =

b

m b

10 ζijk bi aj ak ri−2 rj+1 rk+1

j=0 i=1 k=0 k even j even

+

b

b

m

j=0 i=1 k=0 k even j even i odd

11 ζijk bi aj ak ri−2 rj+1 rk+1 ,

94

Chapter 1. The Averaging Theory for Computing Periodic Orbits

where 10 1 ζijk = δjk i(i − 1)α1j α1k M10 (i, ρ1 , 0), 11 ζijk =

b+2

2 δjkρ i(i − 1)α1j α ρ1 +2 k M11 (i, ρ1 , 0), 1 2

ρ1 =1 ρ1 even

with

) 1 δjk

=

1 if j = k, 2 if j = k,

) and

2 δjkρ 1

=

1 if j = k, ρ1 = 0, 0. 2 if j = k, ρ1 =

2π Thus, H8 (r) = 0 h2 (r)s22 (r)dθ is a polynomial in r of degree m + 2n if n even, and m + 2n − 2 if n odd. From

0 2π

2π

cos2i+1 θ sin θ sin(ρ1 θ)dθ = 0,

cos2i θ sin θ sin(ρ1 θ)dθ = M12 (i, ρ1 , 0) = 0,

0 2π

2π

ρ1 odd, ρ1 odd,

cosi θ(sin θ)θdθ = M13 (i, 0, 0) = 0,

0 2π

cos2i θ sin θ sin(ρ1 θ)dθ = M14 (i, ρ1 , 0) = 0,

cos2i+1 θ sin θ sin(ρ1 θ)dθ = 0,

ρ1 even, ρ1 even,

for i = 1, 2, . . ., we have that

2π

h2 (r)(s1 (r) + s2 (r))g2 (r)dθ = 0

m l m k=0

+

j=1 i=1 j odd

m m b k=0

+

12 ζijk bi aj bk ri−2 rj+1 rk

13 ζijk bi aj bk ri−2 rj+1 rk

j=0 i=1 j even

m m l i=1 k=0 j=1 j even

14 ζijk bi aj bk ri−2 rj+1 rk ,

1.4. Three applications of Theorem 1.3.5

95

where

12 ζijk

⎧ j+2 i(i − 1) ⎪ ⎪ ⎨ α ρ+1 j M12 (i, ρ1 , 0) for k + i even, 2 k+2 = ρ1 =1 ⎪ ρ odd ⎪ 1 ⎩ 0 for k + i odd, i(i − 1) α1j M13 (i, 0, 0), k+1 ⎧ j+2 ⎪ i(i − 1) ⎪ ⎨ α ρ1 +2 j M14 (i, ρ1 , 0) for k + i even, 2 k+2 = ρ1 =0 ⎪ ρ1 even ⎪ ⎩ 0 for k + i odd.

13 = ζijk

14 ζijk

2π Thus, H9 (r) = 0 h2 (r)(s1 (r) + s2 (r))g2 (r)dθ is a polynomial in r of degree 2m + n − 1 if n even, and 2m + n − 2 if n odd. From the value of the integral

2π

cosi θ sin θdθ = 0,

for i = 1, 2, . . ., we have that H10 (r) = We conclude that

2π 0

2π 0

h2 (r)g22 (r)dθ = 0.

1 ∂ 2 F1 (s, r)(y1 (s, r))2 ds = Hi , 2 2 ∂r i=1 10

whose degree is the greatest of the degrees of Hi . Hence, the proof of the lemma follows. The proofs of the next three lemmas follow in a similar way to the previous one; they will be omitted. Lemma 1.4.9. The integral

2π 0

1 ∂F1 2 ∂r (s, r)(y2 (s, r))ds

equals the polynomial

π (E0 + E1 r + E2 r2 + · · · + Eϑ rϑ ), r where

⎧ n + 2m ⎪ ⎪ ⎪ ⎨n + 2m − 1 ϑ= ⎪ 3n + 2 ⎪ ⎪ ⎩ 3n + 1

if if if if

m>n+1 m>n+1 m≤n+1 m≤n+1

and and and and

n n n n

is is is is

(1.163)

even, odd, even, odd,

96

Chapter 1. The Averaging Theory for Computing Periodic Orbits

and

E2l+1 =

i+j+k=2l−1

+

2 ηij ai dj

i+j=2l

+

i+j+k=2l+1

2 δij b i cj

i+j=2l

2 υijk ai aj bk π,

2 βijk ai aj ak +

i+j+k=2l−2

+

2 γijk ai b j b k +

i+j+k=2l i even

E2l =

2 βijk ai aj ak +

i+j+k=2l

2 ηij ai dj +

i+j=2l−1

2 γijk ai b j b k +

2 δij b i cj

i+j=2l−1

2 υijk ai aj b k π +

i+j+k=2l−1 i even

2 ςij ai cj π,

i+j=2l−2 i even

2 2 2 2 2 2 for l = 0, 1, . . . , ϑ/2, where βijk , γijk , δij , ηij , υijk , ςij are real constants. 2π 1 ∂F2 Lemma 1.4.10. The integral 0 2 ∂r (s, r)(y1 (s, r))ds equals the polynomial

π (F0 + F1 r + F2 r2 + · · · + Fν rν ), r where

⎧ n + 2m ⎪ ⎪ ⎪ ⎨n + 2m − 1 ν= ⎪ 3n + 2 ⎪ ⎪ ⎩ 3n + 1

if if if if

m>n+1 m>n+1 m≤n+1 m≤n+1

and and and and

n n n n

(1.164)

is is is is

even, odd, even, odd,

and

F2l+1 =

i+j+k=2l−1

+

3 βijk ai aj ak +

3 γijk ai b j b k +

i+j+k=2l+1

3 δij b i cj

i+j=2l

3 ηij ai dj ,

i+j=2l

F2l =

3 βijk ai aj ak +

i+j+k=2l−2

+

i+j=2l−1

3 γijk ai b j b k +

i+j+k=2l

3 ηij ai dj +

3 υijk ai aj b k π +

i+j+k=2l−1 i even

3 δij b i cj

i+j=2l−1

3 ςij ai cj π,

i+j+=2l−2 i even

3 3 3 3 3 3 for l = 0, 1, . . . , ν/2, where βijk , γijk , δij , ηij , υijk , ςij are real constants. 2π Lemma 1.4.11. The integral 0 F3 (s, r)ds equals the polynomial

π (G0 + G2 r2 + · · · + Gψ rψ ), r

(1.165)

1.4. Three applications of Theorem 1.3.5 where

⎧ n + 2m ⎪ ⎪ ⎪ ⎨n + 2m − 1 ψ= ⎪ 3n + 2 ⎪ ⎪ ⎩ 3n + 1

if if if if

97

m>n+1 m>n+1 m≤n+1 m≤n+1

and and and and

n n n n

is is is is

even, odd, even, odd,

and

G2l =

i+j+k=2l−2

+

4 βijk ai aj ak +

4 γijk ai b j b k +

i+j+k=2l

4 δij b i cj

i+j=2l−1

4 ηij ai dj + p2l−2 ,

i+j=2l−1 4 4 4 2 4 for l = 0, 1, . . . , ψ/2, where βijk , γijk , δij , ηij , υijk are real constants.

By Lemmas 1.4.8, 1.4.9, 1.4.10 and 1.4.11 we obtain F30 (r) =

α M0 + M1 r + M2 r2 + M3 r3 + M4 r4 + · · · + M−1 r−1 + M r , r

where

M2l+1 =

i+j+k=2l−1

+

ηij ai dj +

i+j=2l

M2l =

βijk ai bj bk +

i+j+k=2l+1

ηij ai dj +

νijk ai aj bk +

i+j+k=2l−1 i even

i+j+k=2l−2 i even

τijk ai aj ak π 2 ,

γijk ai aj ak +

δij bi cj

i+j=2l−1

μijk ai aj ak + 2l−2 p2l−2

i+j+k=2l−2

⎛

+

δij bi cj

i+j=2l

νij ai aj bk π,

i+j+k=2l−2

i+j=2l−1

⎜ +⎝

γijk ai bj bk +

i+j=2l i even

i+j+k=2l

+

βijk ai aj ak +

i+j=2l−2 i even

⎞ ⎟ ρijk ai cj ⎠ π

98

Chapter 1. The Averaging Theory for Computing Periodic Orbits

for l = 0, 1, 2, . . . , /2 and ⎧ n + 2m ⎪ ⎪ ⎪ ⎨n + 2m − 1 = ⎪ 3n + 2 ⎪ ⎪ ⎩ 3n + 1

if if if if

m>n+1 m>n+1 m≤n+1 m≤n+1

and and and and

n n n n

is is is is

even, odd, even, odd.

Applying the equalities ai = 0, for all i even and (1.161), we obtain that M0 = 0 and Mκ = 0 for κ odd. Moreover, from (1.161) we obtain ai b j = 0 ck = i+j=k+1 i odd j even

for k > b. Then Mk = 0 for k greater than ⎧ n + m − 2 if n and m are odd, ⎪ ⎪ ⎪ ⎨ n + m − 1 if n is odd and m is even, λ= ⎪ n + m − 2 if n and m are even, ⎪ ⎪ ⎩ n + m − 1 if n is even and m is odd. Thus F30 (r) = αr M2 + M4 r2 + M6 r4 + · · · + Mλ−4 rλ−2 + Mλ−2 rλ , where Mω =

i+j+k=ω i odd j even k odd

βijk ai b j b k +

i+j=ω−1 i even j odd

δij b i cj +

ηij ai dj + ω pω−2 .

i+j=ω−1 i odd j even

Consequently, F3 (z) is a polynomial of degree λ in the variable r2 . Then F3 (z) has at most [(n + m − 1)/2] positive roots and, from the third order averaging method, we conclude that this is the maximum number of limit cycles of the polynomial Li´enard diﬀerential systems (1.152) with k = 3 bifurcating from the periodic orbits of the linear center x˙ = y, y˙ = −x. This completes the proof of statement (iii) of Theorem 1.4.6.

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[73] R. Ortega. “Stability and index of periodic solutions of an equation of Duﬃng type”. Boo. Uni. Mat. Ital B 3 (1989), 533–546. [74] J. Ozaki and S. Kurosaki. “Periodic orbits of H´enon–Heiles Hamiltonian”. Prog. in Theo. Phys. 95 (1996), 519–529. [75] H. Poincar´e. “M´emoire sur les courbes d´eﬁnies par les ´equations diﬀ´erentielles”. Oeuvreus de Henri Poincar´e, Vol. I, Gauthiers–Villars, Paris (1951), 95–114. [76] M. Roseau. “Vibrations non-lin´eaires et th´eorie de la stabilit´e”. Springer Tracts in Natural Philosophy 8, Springer–Verlag, Berlin–New York, 1966. [77] G.S. Rychkov. “The maximum number of limit cycle of the system x˙ = y − a1 x3 − a2 x5 , y˙ = −x is two”. Diﬀerential’nye Uravneniya 11 (1975), 380–391. [78] J.A. Sanders F. Verhulst and J. Murdock. “Averaging Methods in Nonlinear Dynamical Systems” (second edition). Applied Mathematical Sciences 59, Springer, New York, 2007. [79] M. Santoprete. “Block regularization of the Kepler problem on surfaces of revolution with positive constant curvature”. J. Diﬀerential Equations 247 (2009), 1043–1063. [80] I.R. Shafarevich. “Basic Algebraic Geometry”. Springer, 1974. [81] K.S. Sibirskii. “On the number of limit cycles in the neighborhood of a singular point”. Diﬀerential Equations 1 (1965), 36–47. [82] D. Strozzi. “On the origin of interannual and irregular behaviour in the El Ni˜ no properties”. Report of the Department of Physics, Princeton University (1999). ´ [83] G. Swirszcz. “Cyclicity of inﬁnite contour around certain reversible quadratic center”. J. Diﬀerential Equations 154 (1999), 239–266. [84] G.K. Vallis. “Conceptual models of El Ni˜ no and the southern oscillation”. J. Geophys. Res. 93 (1988), 13979–13991. [85] F. Verhulst. “Nonlinear dierential equations and dynamical systems”. Universitext, Springer, 1991. [86] K.N. Webster and J. Elgin. “Asymptotic analysis of the Michelson system”. Nonlinearity 16 (2003), 2149–2162. [87] P. Yu and M. Han. “Limit cycles in generalized Li´enard systems”. Chaos, Solitons and Fractals 30 (2006), 1048–1068. [88] H. Zola¸ dek. “The cyclicity of triangles and segments in quadratic systems”. J. Diﬀerential Equations 122 (1995), 137–159.

Chapter 2

Central Conﬁgurations Richard Moeckel The topic of the present chapter is one of my favorites: central conﬁgurations of the n-body problem. I gave a course on the same subject in Trieste in 1994 and wrote up some notes (by hand) which can be found on my website [23]. For the new course, I tried to focus on some new ideas and techniques which have been developed in the intervening twenty years. In particular, I consider space dimensions bigger than three. There are still a lot of open problems and it remains an attractive area for mathematical research.

2.1 The n-body problem The Newtonian n-body problem is the study of the dynamics of n point particles with masses mi > 0 and positions xi ∈ Rd , moving according to Newton’s laws of motion: mi mj (xi − xj ) , 1 ≤ j ≤ n, (2.1) mj x¨j = 3 rij i =j

where rij = |xi − xj | is the Euclidean distance between xi and xj . Although we are mainly interested in dimensions d ≤ 3, it is illuminating and entertaining to consider higher dimensions as well. Let x = (x1 , . . . , xn ) ∈ Rdn be the conﬁguration vector and let mi mj U (x) = (2.2) rij i 0 is constant, then x ˜(t) = λ2 x(λ−3 t) is also a solution. This will be called the scaling symmetry of the n-body problem. For any conﬁguration x, the moment of inertia with respect to the center of mass is I(x) = (x − c)T M (x − c) = mj |xj − c|2 , (2.8) j

where y is the corresponding centered conﬁguration. I(x) is homogeneous of degree 2 with respect to the scaling symmetry. The following alternative formula in terms of mutual distances is also useful: I(x) =

1 2 mi mj rij . m0 i 0, I(x)

where I(x) is the moment of inertia with respect to c from (2.8). If x is a central conﬁguration then the gravitational acceleration on the j-th body due to the other bodies is x ¨j = m1j ∇j U (x) = −λ(xj − c). In other words, all of the accelerations are pointing towards the center of mass, c, and are proportional to the distance from c. We will see that this delicate balancing of the gravitational forces gives rise to some remarkably simple solutions of the n-body problem. Before describing some of these, we will brieﬂy consider the question of existence of central conﬁgurations. For given masses m1 , . . . , mn , it is far from clear that (2.10) has any solutions at all. We will consider this question in due course. For now we just note the existence of symmetrical examples for equal masses. If all n masses are equal we can arrange the bodies at the vertices of a regular polygon, polyhedron or polytope. Then it follows from symmetry that the acceleration vectors of each mass must point toward the barycenter of the conﬁguration. This is the condition for a central conﬁguration, i.e., there will be some λ for which the CC equations hold. In R2 we can put three equal masses at the vertices of an equilateral triangle or n equal masses at the vertices of a regular n-gon to get simple examples. One can also put an arbitrary mass at the center of a regular n-gon of equal masses as in Figure 2.1 (left). In R3 we have the ﬁve regular Platonic solids, the tetrahedron, cube, octahedron, dodecahedron and icosahedron. It is not clear what to do if n = 4, 6, 8, 12, 20, however. It turns out that there are six kinds of regular, convex four-dimensional polytopes but in higher dimensions there are only three, namely the obvious generalization of the tetrahedron, cube and octahedron [9, 16]. The regular d-simplex provides an example of a central conﬁguration of d + 1 equal masses in Rd generalizing the equilateral triangle and tetrahedron. Remarkably, these turn out to be central conﬁgurations even when the masses are not equal (see Proposition 2.8.6) so we do indeed have at least one CC for any choice of masses, provided we are willing to work in high-dimensional spaces. As a special case, note that for the two-body problem, every conﬁguration is a regular simplex, i.e., a line segment. So every conﬁguration of n = 2 bodies is a central conﬁguration.

110

Chapter 2. Central Conﬁgurations

Less obvious examples can be found by numerically solving (2.10), for example the asymmetrical CC of 8 equal masses shown in Figure 2.1 (right).

Figure 2.1: Central conﬁgurations. Central conﬁgurations can be used to construct simple, special solutions of the n-body problem where the shape of the ﬁgure formed by the bodies remains constant. The conﬁguration changes only by simultaneous translation, rotation and scaling. In other words, the conﬁgurations x(t) at diﬀerent times are all similar. In this case the conﬁguration relative to the center of mass will change only by scaling and rotation. Deﬁnition 2.3.2. A solution of the n-body problem is self-similar or homographic if it satisﬁes x(t) − c(t) = r(t)Q(t)(x0 − c0 ), (2.11) where x0 is a constant conﬁguration, r(t) > 0 is a real scaling factor, and Q(t) ∈ SO(d) is a rotation. Here c(t), c0 are the centers of mass of x(t), x0 . Two special cases are the homothetic solutions, where x(t) − c(t) = r(t)(x0 − c0 ),

(2.12)

and the rigid motions or relative equilibrium solutions, where x(t) − c(t) = Q(t)(x0 − c0 ).

(2.13)

The simplest of these are the homothetic solutions. For example, if put three equal masses at the vertices of an equilateral triangle and release them with initial velocities all zero, it seems clear that the triangle will just collapse to the center of mass with each particle just moving on a line towards the center. It turns out that such a solution is possible only when x0 is a central conﬁguration. Proposition 2.3.3. If x0 is a central conﬁguration with constant λ and if r(t) is any solution of the one-dimensional Kepler problem r¨(t) = −

λ , r(t)2

(2.14)

then x(t) as in (2.12) is a homothetic solution of the n-body problem, and every homothetic solution is of this form.

2.3. Central conﬁgurations and self-similar solutions

111

Proof. Substituting x(t) from (2.12) into Newton’s equation (2.4) gives r¨(t)M (x0 − c0 ) = ∇U (x(t)) = r(t)−2 ∇U (x0 ). Now ∇U (x0 ) = 0 for all x0 , so this equation is satisﬁed if and only if there is some constant, call it −λ, such that r¨(t)r(t)2 = −λ,

−λM (x0 − c0 ) = ∇U (x0 ).

The one-dimensional Kepler problem (2.14) describes the motion of a point on a line gravitationally attracted to a mass λ at the origin. It is easy to see qualitatively what will happen even without solving it. For example, the solution r(t) with initial velocity r(0) ˙ = 0 collapses to the origin in both forward and backward time. The corresponding homothetic solutions maintain the shape of the underlying central conﬁguration x0 while collapsing to a total collision at the center of mass in both forward and backward time (see Figure 2.2 for the forwardtime half). Each body moves along a straight line towards the collision. From the examples of central conﬁgurations mentioned above we see that we can have homothetically collapsing solutions in the shape of an equilateral triangle, regular n-gon or regular polytope.

Figure 2.2: The forward-time half of a homothetic solution based on Lagrange’s equilateral triangle with masses 10, 2, and 1. Released with zero velocity, the masses collapse to the center of mass (indicated by the + symbol) along straight lines, maintaining the equilateral shape. It turns out that central conﬁgurations also lead to rigid motions and more general homographic solutions. We will postpone a general discussion of homographic solutions in Rd to later sections. For now we will consider the case of planar motions. Let d = 2 and suppose x0 ∈ R2n is a central conﬁguration. Let cos(θ) − sin(θ) ∈ SO(2). Q(θ) = sin(θ) cos(θ) The most general planar homographic motion would be of the form x(t) − c(t) = r(t)Q(θ(t))(x0 − c0 )

(2.15)

112

Chapter 2. Central Conﬁgurations

for some functions r(t) > 0, θ(t). Substituting this into Newton’s equation leads, after some simpliﬁcations, to ˙ (¨ r − r θ˙2 )M (x0 − c0 ) + (r θ¨ + 2 r˙ θ)JM (x0 − c0 ) = r−2 ∇U (x0 ), where J is the 2n × 2n matrix J = diag(j, . . . , j),

j = Q(θ)−1 Q (θ) =

0 1

−1 . 0

Now (x0 − c0 ) and J(x0 − c0 ) are nonzero, orthogonal vectors in R2n and the latter is also orthogonal to ∇U (x0 ). Therefore, there must be some constant −λ such that ˙ 2=− λ , r¨(t) − r(t)θ(t) r(t)2 (2.16) ¨ + 2r(t) ˙ = 0, r(t)θ(t) ˙ θ(t) and −λM (x0 − c0 ) = ∇U (x0 ). The diﬀerential equation is just the two-dimensional Kepler problem in polar coordinates whose solutions are of the familiar elliptical, parabolic or hyperbolic types and the last equation is the CC equation. Proposition 2.3.4. If x0 is a planar central conﬁguration with constant λ and if r(t), θ(t) is any solution of the two-dimensional Kepler problem (2.16), then (2.15) is a planar homographic solution and every such solution is of this form. As a special case, we could take a circular solution of the Kepler problem with r(t) = 1. Then we get a rigid motion or relative equilibrium solution where the planar central conﬁguration just rotates at constant angular speed around the center of mass. This is the most general relative equilibrium solution in the plane. In particular, nonuniform rotations are not possible. In higher dimensions, the situation regarding rigid solutions and nonhomothetic homographic solutions is more complicated, mainly due to the increased complexity of the rotation group SO(d). The next few sections describe an approach to the general case developed by Albouy and Chenciner.

2.4 Matrix equations of motion We will now describe an interesting reformulation of the n-body problem due to Albouy and Chenciner [2, 3, 7] which is very convenient for studying symmetric solutions. Let X = x1 | · · · |xn , V = v1 | · · · |vn ,

2.4. Matrix equations of motion

113

Figure 2.3: Planar homographic motions based on a central conﬁguration of eight equal masses from Figure 2.1. On the left is a relative equilibrium solution while the solution on the right features elliptical orbits of eccentricity 0.8. be the d × n matrix whose columns are the positions and velocities of the n bodies. For example, the matrix ⎡ ⎤ 1 − 12 − 21 √ √ ⎢ ⎥ ⎢0 23 − 23 ⎥ X =⎢ (2.17) ⎥ ⎣0 0 0 ⎦ 0 0 0 represents a conﬁguration of n = 3 bodies in d = 4 dimensions arranged at the vertices of an equilateral triangle. We will view X, V as linear maps X, V : Rn → Rd . The domain of these maps has no particular physical meaning; it is just a space of n × 1 column vectors ξ with one coordinate for each of the n-bodies. We can think of the standard basis vectors e1 , . . . , en as representing the diﬀerent bodies. While the columns of X, V have an immediate dynamical meaning, it is not clear what to think about the rows. These are 1 × n vectors which we will view as n∗ elements of the dual space R , another nonphysical space. For example, the ﬁrst 1 1 row 1 − 2 − 2 of the matrix above gives the coeﬃcients of a linear function whose values on the basis vectors e1 , e2 , e3 of R3 are the ﬁrst coordinates of the three bodies in R4 . To get the matrix version of the laws of motion, write the j-th acceleration vector from Newton’s equations (2.1) as a linear combination of the position vectors: ⎛ ⎞ mi (xi − xj ) mi mi ⎠ 1 x¨j = ∇j U (x) = = xi 3 − xj ⎝ . 3 3 mj rij rij rij i =j

i =j

i =j

114

Chapter 2. Central Conﬁgurations

So we get the matrix equation: ¨ = XA(X), X

(2.18)

where A(X) is the n × n matrix ⎡

A11

⎢ m2 ⎢ r3 12 A(X) = ⎢ ⎢ .. ⎣ .

mn 3 r1n

m1 3 r12

A22 mn 3 r2n

··· ··· ···

m1 3 r1n m2 3 r2n

⎤

⎥ ⎥ ⎥ .. ⎥ , . ⎦

Ajj = −

Aij = −

i =j

mi i =j

3 rij

.

(2.19)

Ann

Note that A(X) is invariant under translations and rotations, since it involves only the mutual distances. It is independent of the space dimension d. For example, consider the three-body problem in Rd where we have the 3 × 3 matrix ⎡ m2 ⎤ m1 m1 3 − r3 − m 3 3 3 r13 r12 r13 12 ⎢ ⎥ ⎢ ⎥ m2 m2 − rm31 − rm33 A=⎢ 3 3 ⎥. r12 r 12 23 23 ⎣ ⎦ m3 m3 m1 m2 − r3 − r3 r3 r3 13

23

13

23

A(X) has some other useful properties. Let M = diag(m1 , . . . , mn ) be an n × n version of the mass matrix. Then we have XA(X)M = ∇1 U (X) · · · ∇n U (X) . In addition, A(X)M is symmetric: AM = (AM )T = M AT . Finally, A(X)M is negative semi-deﬁnite. Indeed, for any ξ ∈ Rn one can check that mi mj (ξi − ξj )2 . ξ T AM ξ = − 3 r ij i 0 one can check that the eigenvalue ratio σ22 /σ12 of S varies over ((1 + 4m3 )/(8 + 4m3 ), ∞) as the angle at m3 of the isosceles shape decreases from π to 0.

2.7 Homographic motions in Rd Next we will show that the orbits described in Proposition 2.5.1 are actually the most general, nonrigid homographic motions. In particular, only central conﬁgurations give rise to such motions. Proposition 2.7.1. Every nonrigid homographic solution of the n-body problem in Rd is of the form X(t) − C(t) = r(t)Q(t)(X0 − C0 ),

Q(t) = exp(θ(t)J),

where X0 is a central conﬁguration with constant λ, (r(t), θ(t)) is a solution of the Kepler problem (2.16), and J is an antisymmetric d × d matrix with J 2 |C(X0 ) = −I|C(X0 ) . Proof following [7]. Since the motion is homographic, the right-hand side of equation (2.18) is X(t)A(X(t)) = r(t)−3 X(t)A(X0 ). The fact that the n × n matrix A(X0 ) is M −1 -symmetric implies that it is diagonalizable. One of the eigenvalues is zero since the mass vector m is in the kernel, and the others are nonpositive because of the negative semi-deﬁniteness of AM . Let R be an invertible n × n matrix with R−1 A(X0 )R = diag(−λ1 , −λ2 , . . . , −λn ). If W (t) = (X(t) − C(t))R then Newton’s equations give ¨ = r(t)−3 X(t)A(X0 )R = r(t)−3 W (t)R−1 A(X0 )R W and so the columns wj (t) of W (t) satisfy w ¨j (t) = −

λj wj (t) . r(t)3

128

Chapter 2. Central Conﬁgurations

Since the solution is homographic, we have W (t) = r(t)Q(t)W0 where W0 = (X0 − C0 )R. It follows that the columns of W, W0 satisfy |wj (t)| = r(t)|w0j |,

j = 1, . . . , n.

For each column such that |w0j | = 0, deﬁne uj (t) = wj (t)/|w0j |. Then |uj (t)| = r(t) for j = 1, . . . , n and λj uj (t) , u¨j (t) = − |uj (t)|3 i.e., the normalized nonzero columns solve Kepler’s equations with constant λj . Moreover, they all have the same norm r(t). It follows that each of these uj (t) moves in a plane and can be represented with respect to polar coordinates in that plane by functions r(t), θ(t) satisfying (2.16) with λ = λj . ˙ Lemma 2.7.2. If r(t), θ(t) solves (2.16) and r(t) is not constant, then λ and θ(t) are uniquely determined by r(t).

Proof. Exercise.

Continuing with the proof of the proposition, we now see that all of the λj corresponding to nonzero columns of W (t) are equal. Then we have X0 A(X0 ) = W0 diag(−λ1 , . . . , −λn )S −1 = −λW0 S −1 = −λ(X0 − C0 ), where the second equation holds because changing λj to λ for a column wj = 0 does no harm. This shows that X0 is a central conﬁguration. To get the rest we will use the reduced equations of motion (2.28). Since we are assuming that X(t) is homographic, the relative state matrices have a particularly simple form. Let Y (t) = X(t) − C(t) = X(t)P and W (t) = V (t)P be the centered position and velocity matrices. Then Y (t) = r(t)Q(t)Y0 and W (t) = ˙ r(t)Q(t)Y ˙ 0 + r(t)Q(t)Y0 . The relative state matrices are B(t) = r(t)2 B0 ,

C(t) = r(t)r(t)B ˙ 0,

D(t) = r(t) ˙ 2 B0 − r(t)2 Y0T Ω(t)2 Y0 ,

˙ ∈ so(d). The antisymmetry of this matrix implies that where Ω(t) = Q(t)T Q(t) terms involving Y0T Ω(t)Y0 in the calculation of these matrices vanish. Now calcu˙ lating C(t) and comparing with (2.28) gives (r¨ r + r˙ 2 )B0 = D + BA = D − λr2 B0 ,

(2.40)

where we used (2.29). Now we already found that r(t), θ(t) are solutions of Kepler’s equation. By rescaling X0 and choosing the origin of time, we may assume that r(0) = 1 and r(0) ˙ = 0. The second assumption certainly holds at the perihelion of the Kepler orbit. At this point the velocities and positions are orthogonal. Evaluating (2.40) at t = 0 and using Kepler’s equation (2.16) we get D0 = θ˙02 B0 .

(2.41)

2.8. Central conﬁgurations as critical points

129

We also have C0 = 0. Let Z0 = Y0 W0 be the initial state matrix and consider the matrices L 1 = Z0 , !

We have LT1 L2

θ˙0−1 C0 = −1 θ˙0 D0

L2 = θ˙0−1 W0

−θ˙0 Y0 .

" ! 0 −θ˙0 B0 = −1 ˙ ˙ θ0 D 0 −θ0 C0

" −θ˙0 B0 . 0

This 2n × 2n matrix is antisymmetric by (2.41) so, by Lemma 2.6.5, there is an antisymmetric d × d matrix J such that W0 = θ˙0 Jy0 ,

Y0 = −θ˙0−1 W0 = −J 2 Y0 .

By Proposition 2.5.1, Y˜ (t) = exp(θ(t)J)Y0 is a homographic solution and its initial conditions ˜ (0) = θ˙0 JY0 = W0 Y˜ (0) = Y0 , W are the same as those of the given homographic solution, Y (t). Therefore, Y (t) = exp(θ(t)J)Y0 as claimed. Although we have made a point of studying the special solutions of the nbody problem in Rd , we will summarize the results for the physical case d = 3. The homographic solutions in R3 are of the following types. For any central conﬁguration and any solution of the one-dimensional Kepler problem there is a homothetic solution. For any central conﬁguration which is contained in some two-dimensional subspace and any solution of the two-dimensional Kepler problem, there is a homographic solution for which the bodies remain in the same plane. This is a uniform planar rigid motion if we take the circular solution of the Kepler problem. There are no other homographic motions. In particular, a nonplanar CC does not lead to any rigid or homographic, nonhomothetic solutions. A conﬁguration which is balanced but not central is not balanced in R3 so does not give rise to a RE solution in R3 .

2.8 Central conﬁgurations as critical points Now that we have some motivation for studying central conﬁgurations, lots of interesting questions arise. Fixing the masses mi we can ask whether central conﬁgurations exist and if so, how many there are up to symmetry. Working with conﬁguration vectors x ∈ Rdn we need to study solutions of the CC equation ∇U (x) + λM (x − c) = 0.

(2.42)

If x is a CC then so is any conﬁguration y obtained from x by translations and rotations. In particular, the centered conﬁguration x − c is also a CC. If k > 0

130

Chapter 2. Central Conﬁgurations

then kx is also a central conﬁguration but with a diﬀerent λ. Recall that λ(x) = U (x)/I(x), where I(x) is the moment of inertia around the center of mass. So λ(kx) = λ(x)/k 3 . We will view such CC’s as equivalent and refer to similarity classes of CC’s. The key idea in this section is to interpret CC’s as constrained critical points of the Newtonian potential. The constraint is just to ﬁx the moment of inertia. Since ∇I(x) = 2M (x − c), the CC equation can be written ∇U (x) + 12 λ∇I(x) = 0. Interpreting λ/2 as Lagrange multiplier, we get: Proposition 2.8.1. A conﬁguration vector x0 is a central conﬁguration if and only if it is a critical point of U (x) subject to the constraint I(x) = k, where k = I(x0 ). It is useful for existence proofs to have a compact constraint set. We can use the scaling symmetry to normalize the moment of inertia to be I(x) = 1 but, because of the translation invariance, {x : I(x) = 1} is not compact. We can eliminate the translation symmetry by ﬁxing the center of mass. Just as in the matrix formulation of the problem, we can view the passage from x to x − c as an orthogonal projection. In fact x − c = Pˆ x, where Pˆ : Rdn → Rdn is the orthogonal projection onto the subspace where m1 x1 + · · · + mn xn = 0 ∈ Rd with respect to the mass inner product v T M w. The matrix of Pˆ is 1 ˆT ˆ ˆ = I I ··· I , Pˆ = I − L LM, L (2.43) m0 ˆ is d × dn with blocks of d × d identity matrices. One can check that Pˆ is where L an M -symmetric projection matrix. Deﬁne the normalized conﬁguration space as ˆ x = 0, I(x) = 1}. N = {x : c = LM Any conﬁguration x determines a unique normalized conﬁguration with c = 0 and I = 1. Note that the center of mass condition deﬁnes a subspace of Rdn of dimension d(n − 1) and then I = 1 gives an ellipsoid in this subspace. Hence N is a smooth compact manifold diﬀeomorphic to a sphere, N Sd(n−1)−1 . Proposition 2.8.2. A conﬁguration vector x is a central conﬁguration if and only if the corresponding normalized conﬁguration is a critical point of the Newtonian potential U (x) restricted to N .

2.8. Central conﬁgurations as critical points

131

Proof. If x is a CC, so is the corresponding normalized conﬁguration. Proposition 2.8.1 shows that this normalized conﬁguration is a critical point of U (x) with the constraint I(x) = 1, so it is still a critical point if we add the center of mass constraint deﬁning N . Conversely, suppose x is a critical point of U (x) restricted to N . We need to show that it is still a critical point if we remove the center of mass constraint. This can be checked using the orthogonal projection Pˆ . Note that N is a smooth ˆ ⊂ Rdn . Therefore x ∈ N is codimension one submanifold of the subspace ker LM a critical point of U |N if and only if (DU (x) + kDIS (x)) v = 0 ˆ , where k ∈ R is a Lagrange multiplier. Equivalently we need for all v ∈ ker LM (DU (x) + kDIS (x)) Pˆ = 0, ˆ where Pˆ is the orthogonal projection onto ker LM from (2.43). By translation ˆ invariance U (P x) = U (x), and diﬀerentiation gives DU (x)Pˆ = DU (x) for x ∈ N . Similarly, DI(x)Pˆ = DI(x). So we can drop Pˆ from the last equation and take transposes to get ∇U (x) + k ∇I(x) = 0,

which is the CC equation.

An alternative approach is based on the moment of inertia with respect to the origin, n T mj |xj |2 . I0 (x) = x M x = j=1

For conﬁgurations with c = 0, I(x) = I0 (x) and the CC equation becomes ∇U (x) + λM x = 0.

(2.44)

This is the critical point equation with ﬁxed I0 (x). It turns out that (2.44) forces c = 0 and we have: Proposition 2.8.3. The point x is a critical point of U (x) on {x : I0 (x) = 1} if and only if x is a normalized central conﬁguration. Proof. If x ∈ N then c = 0 and I(x) = I0 (x) = 1. If it is also a central conﬁguration then (2.44) holds, so it is a critical point of U (x) on {I0 = 1}. Conversely, suppose x is a critical point of U (x) on {I0 = 1}. Then (2.44) holds. We will show that this implies c = 0 and it follows that x ∈ N and that the CC equation (2.42) holds. Equation (2.44) gives λmj xj = −∇j U (x) = Fji , i =j

132

Chapter 2. Central Conﬁgurations

3 where Fji = (mi mj (xi − xj ))/rij is the force on body j due to body i. Summing over j and dividing by the total mass gives

λc =

1 Fij . m0 i 0 we get c = 0 as required. The manifold {x : I0 (x) = 1} is diﬀeomorphic to the sphere Sdn−1 so this approach gives compactness without explicitly imposing the center of mass constraint. The critical points will automatically lie in our previous constraint manifold N . It is also possible to treat balanced conﬁgurations as critical points. Modify the vector version of the balance equation (2.37) by introducing a constant λ to get ˆ (x − c) = 0. ∇U (x) + λSM (2.45) Here λ ∈ R and Sˆ = diag(S, . . . , S) is a dn × dn block-diagonal matrix with identical d × d blocks S, the positive semi-deﬁnite, symmetric matrix from Proposition 2.6.7. We will call x an S-balanced conﬁguration (SBC) if (2.45) holds for some λ. CC’s are a special case with S = I. By putting a λ into (2.45) we can say that x and kx are both S-balanced. The equation is also invariant under translations but generally not invariant under rotations. In fact, the matrix S transforms under rotations and scalings via S(kQx) = k −3 QSQT . In the CC case we have S = I and we get rotation invariance. The other extreme would be that S has d distinct eigenvalues and then it is not stabilized by any rotation. By choosing an appropriate rotation Q we can get QSQT = diag(σ12 , σ22 , . . . , σd2 ). It is no loss of generality to assume S positive deﬁnite since it is deﬁnite on C(x) and we could extend it arbitrarily on C(x)⊥ . To handle SBC’s in a similar way to CC’s, we will deﬁne an S-weighted moment of inertia. Assuming that S is positive deﬁnite, we can use it to deﬁne a new inner product and norm on Rd , ξ, η S = ξ T Sη,

|ξ|2S = ξ T Sξ.

Then set ˆ (x − c) = IS (x) = (x − c)T SM

n j=1

mj |xj − c|2S .

2.8. Central conﬁgurations as critical points

133

As in the CC case, the constant λ in (2.45) is λ = U (x)/IS (x). Deﬁne the Snormalized conﬁguration space ˆ x = 0, IS (x) = 1}. N (S) = {x : c = LM Then, as for CC’s, we have: Proposition 2.8.4. A conﬁguration vector x is a S-balanced conﬁguration if and only if the corresponding normalized conﬁguration is a critical point of U (x) restricted to N (S). One of the main applications of the characterization of CC’s and SBC’s as critical points are the existence proofs. For example: Corollary 2.8.5. For every choice of masses mi > 0 in the n-body problem in Rd , there is at least one central conﬁguration. For every choice of masses and every d × d positive deﬁnite symmetric matrix S, there exists at least one S-balanced conﬁguration. Proof. It suﬃces to consider SBC’s, since CC’s are a special case. Note that N (S) is a compact submanifold of Rdn . The Newtonian potential deﬁnes a smooth function U : N (S) \ Δ → R. The singular set N (S) ∩ Δ is compact and U (x) → ∞ as x → Δ. It follows that U attains a minimum at some point x ∈ N (S) \ Δ and this point will be an S-balanced conﬁguration. Although restricting to the compact space N or N (S) is useful, there are a couple of alternative variational characterizations of CC’s and SBC’s as unconstrained critical points. The ﬁrst version is obtained by normalizing the constant λ instead of the moment of inertia. For every solution of (2.42) or (2.45), there is a rescaled solution with λ = k, where k > 0 is any positive constant. If we choose k = 2 then this rescaled conﬁguration will be a critical point of the function F (x) = U (x) + IS (x) on Rdn , i.e., with no constraint on x. Or, we can impose the linear constraint c = 0. Another variational approach is to avoid normalization altogether and look for critical points of the homogeneous function G(x) = IS (x) U (x) or IS (x)U (x)2 . One can check that if x is a solution of (2.45) we get a ray of critical points kx, k > 0, for these functions. In the CC case, the Newtonian potential determines a function on the quotient space M = (N \ Δ)/ SO(d). However, for d > 2 the action of the rotation group is not free and the quotient space is not a manifold. We can get a manifold by restricting to the conﬁgurations of a given dimension.

134

Chapter 2. Central Conﬁgurations

An amusing application of the variational approach on a reduced space is the study of central conﬁgurations of maximal dimension. For any conﬁguration of n-bodies, the centered position space has dim C(x) ≤ n − 1. We will look for CC’s with dim C(x) = n − 1. Proposition 2.8.6. The only central conﬁguration of n-bodies with dim C(x) = n−1 is the regular n-simplex and it is a central conﬁguration for all choices of the masses. Proof. Without loss of generality we can consider the n-body problem in Rn−1 . The conﬁguration space is Rn(n−1) \ Δ and the centered conﬁgurations form a subspace of dimension n(n − 1) − (n − 1) = (n − 1)2 . The subset of conﬁgurations with dim C(x) = n − 1 is an open subset. The rotation group SO(n − 1) acts freely on this open set and we can look for critical points on the quotient space which will be a smooth manifold of dimension (n − 1)2 −

n(n − 1) (n − 1)(n − 2) = . 2 2

The dimension suggests using the mutual distances rij , 1 ≤ i < j ≤ n, as local coordinates. We will look for unconstrained critical points of U (x) + I(x), where we express both terms as functions of the rij using (2.2) and (2.9). We get ∂U ∂I mi mj 2mi mj rij + =− 2 + = 0. ∂rij ∂rij rij m0 3 = m0 /2. The masses cancel out and the mutual distances are equal to rij

The variational characterization suggests using the gradient ﬂow of the Newtonian potential to understand central or balanced conﬁgurations. Generically, a smooth function on a smooth manifold is a Morse function, i.e., it has isolated critical points which are nondegenerate. Due to the rotational symmetry, critical points of U |N will never be isolated for d ≥ 2. One can try to eliminate the rotational symmetry or just work with the similarity classes of critical points. We can still hope for these classes to be isolated from one another or nondegenerate in some sense. First we deal with another problematic aspect of the gradient ﬂow, the lack of compactness. The manifold N (S) is compact, but the ﬂow is only deﬁned on the open subset N (S) \ Δ. The next result, known as Shub’s lemma [32], shows that CC’s and SBC’s are bounded away from Δ. Proposition 2.8.7. For ﬁxed masses m1 , . . . , mn and a ﬁxed positive deﬁnite symmetric matrix S, there is a neighborhood of Δ in N (S) which contains no Sbalanced conﬁgurations. Proof. Otherwise, there would be some x ¯ ∈ N (S) ∩ Δ and a sequence of SBC’s xk ∈ N (S) with xk → x ¯ as k → ∞. The collision conﬁguration x ¯ deﬁnes a partition of the bodies into clusters, where mi , mj are in the same cluster if x ¯i = x ¯j . For k

2.8. Central conﬁgurations as critical points

135

large, the bodies in each cluster will be close to each other but the clusters will be bounded away from one another. Let Fi (xk ) = ∇i U (xk ) be the force on the i-th body. Since xk is a normalized SBC, we have Fi = −λk mi Sxki , λk = U (xk ). Let γ ⊂ {1, . . . , n} be the set of subscripts of one of the clusters. Then, Fi = −λk S mi xki . i∈γ

(2.46)

i∈γ

As k → ∞, we have λk = U (xk ) → ∞ since x¯ ∈ Δ. On the other hand mi xki → Smγ x ¯γ , S i∈γ

where mγ is the total mass of the cluster and x¯γ is the common value of the limiting positions x ¯i , i ∈ γ. We will show below that the left-hand side of (2.46) is bounded. It follows that we must have x ¯γ = 0 for all of the clusters. In other words, there could be only one cluster and it would have to be at the origin. But x) = 1. this is impossible since IS (¯ To see that the left-hand side of (2.46) is bounded, we can split the sum as Fi = Fij + Fil , i∈γ

i,j∈γ i=j

i∈γ l∈γ /

3 where Fij = (mi mj (xj − xi ))/rij is the force on body i due to body j. The ﬁrst sum is identically zero since Fij = −Fji , and the second is bounded by deﬁnition of cluster.

It follows from Shub’s lemma that if the similarity classes of CC’s or SBC’s are isolated then there are only ﬁnitely many of them. To see this, let U denote a neighborhood of Δ in N (S) which contains no SBC’s. Since the complement N (S) \ U is compact, a hypothetical inﬁnite sequence of distinct, similarity classes would have normalized representatives with a convergent subsequence. The limiting conﬁguration would be a nonisolated SBC. If we allow the masses to vary, it is possible to ﬁnd a sequence of CC’s, say x ¯k , converging to Δ. This idea was used by Xia in [36], and further explored in [21]. The masses in each nontrivial cluster all tend to zero. The limiting shapes of the clusters are governed by equations similar to the CC equation. It is interesting to classify CC’s and SBC’s by their Morse index. Recall that if x is a critical point of a smooth function V on a manifold N , there is a Hessian quadratic form on the tangent space Tx N which is given in local coordinates by the symmetric matrix of second partial derivatives, H(x)(v) = v T D2 V (x)v.

136

Chapter 2. Central Conﬁgurations

Alternatively, if γ(t) is any smooth curve in N with γ(0) = x and γ (0) = v then H(x)(v) =

1 d2 V (γ(t))|t=0 . 2 dt2

The Morse index ind(x) is the maximum dimension of a subspace of Tx N on which H(x) is negative deﬁnite. The nullity is the dimension of ker H(x) = {v : H(q)(v, w) = 0 for all w ∈ Tx N }, where H(x)(v, w) = v T D2 V (x)w is the symmetric bilinear form associated to H(x). We are interested in the function V = U |N (S) given by restricting the Newtonian potential to the normalized conﬁguration space. Instead of working in local coordinates, we want to represent the Hessian by a dn × dn matrix, also called H(x), whose restriction to Tx N (S) gives the correct values. Proposition 2.8.8. The Hessian of V : N (S) → R at a critical point x is given by H(x)(v) = v T H(x)v, where H(x) is the dn × dn matrix ˆ H(x) = D2 U (x) + U (x)SM.

(2.47)

Proof. A critical point of V is also an unconstrained critical point of G(x) = IS (x)U (x) in Rdn . Since G|N (S) = U |N (S) = V , their Hessians on Tx N (S) agree. ˆ Pˆ x. For any vector To calculate D2 G ﬁrst recall that IS (x) = xT Pˆ T SM dn w ∈ R , we have ˆ Pˆ w. DIS (x)w = 2xT Pˆ T SM Hence,

1

1

ˆ Pˆ w. DG(x)w = IS (x) 2 DU (x)w + IS (x)− 2 U (x)xT Pˆ T SM

We are only interested in computing D2 G(x)(v, w) where v, w ∈ Tx N (S). In that case we have IS (x) = 1,

Pˆ v = v,

ˆ Pˆ v = 0, xT Pˆ T SM

and analogous equations for w. Diﬀerentiating G again and using these equations we get ˆ w, D2 G(x)(v, w) = D2 U (x)(v, w) + U (x)v T SM as claimed.

It is straightforward to calculate the dn × dn matrix D2 U (x) with the result ⎡ ⎤ D11 D12 · · · D1n ⎢ ⎥ D2 U (x) = ⎣D21 D12 · · · D2n ⎦ , (2.48) .. .. . .

2.8. Central conﬁgurations as critical points

137

where the d × d blocks are Dij =

mi mj I − 3uij uTij , 3 rij

and Dii = −

uij =

xi − xj , rij

for i = j,

Dij .

j =i

The following formula for the value of the Hessian quadratic form on a vector v ∈ Rdn is sometimes useful: mi mj (2.49) −|vij |2 + 3(uij · vij )2 + U (q)v T M v, H(x)(v, v) = 3 rij i 0 any isosceles triangle is balanced with the eigenvalues of S varying with the shape. One can check using a computer that generic choices of isosceles shape lead to nondegenerate SBC’s. In all cases, it is natural to call a critical point nondegenerate if its nullity is as small as possible given the rotational symmetry.

138

Chapter 2. Central Conﬁgurations

Deﬁnition 2.8.10. A CC or SBC in Rd is nondegenerate if the nullity of the corresponding critical point is as small as possible consistent with the rotational symmetry. For CC’s this means that equality should hold in (2.50). For example in R3 a nondegenerate collinear CC has nullity 2, while nondegenerate planar and spatial CC’s have nullity 3.

2.9 Collinear central conﬁgurations The ﬁrst central conﬁgurations were discover by Euler in 1767, see [11]. He studied the collinear three-body problem where he found collinear central conﬁgurations and the corresponding homothetic motions. Moulton investigated the central conﬁgurations of the collinear n-body problem in 1910, see [25]. The results are deﬁnitive in contrast to the state of the theory for d ≥ 2. This section is devoted to proving Moulton’s theorem: Proposition 2.9.1 (Moulton’s Theorem). Given masses mi > 0, there is a unique normalized collinear central conﬁguration for each ordering of the masses along the line. Note that when d = 1 there is no diﬀerence between CC’s and SBC’s due to the lack of variety in 1 × 1 symmetric matrices. It is instructive to start with Euler’s case n = 3. The normalized conﬁguration space N = {x ∈ R3 : m1 x1 + m2 x2 + m3 x3 = 0, m1 x21 + m2 x22 + m3 x23 = m0 } is the curve of intersection of a plane and an ellipsoid. The collision set consists of three planes: Δ = {x1 = x2 } ∪ {x1 = x3 } ∪ {x2 = x3 } which divide the curve into six arcs corresponding to the diﬀerent orderings of the three masses along the line (see Figure 2.6). Since U → ∞ at these points, there must be at least one critical point in each of the arcs. To see that there is only one requires more work. The three mutual distances provide convenient coordinates, but we need to subject them to a collinearity constraint. If we ﬁx the ordering of the bodies to be x1 < x2 < x3 then the constraint is r12 + r23 − r13 = 0. Looking for critical points of the homogeneous function F = U (rij )2 I(rij ) with this constraint, and then normalizing by setting r12 = r, r13 = 1, r23 = 1 − r gives a degree ﬁve polynomial equation for r: (m2 + m3 )r5 + (2m2 + 3m3 )r4 + (m2 + 3m3 )r3 − (3m1 + m2 )r2 − (3m1 + 2m2 )r − (m1 + m2 ) = 0.

(2.51)

Fortunately there is a single sign change so Descartes’ rule of signs implies there is a unique positive real root. Of course, there is no simple formula for how this

2.9. Collinear central conﬁgurations

139 132

312

123

321

213

231

Figure 2.6: N for the collinear three-body problem is the boundary circle of the shaded disk which represents the set I ≤ 1 in the plane of centered conﬁgurations. Δ intersects this plane in three lines which divide the circle into six arcs, one for each ordering of the bodies along the line. root changes as a function of the masses. Euler’s example is a shot over the bow about the CC equation. Even in the simplest nontrivial case, ﬁnding CC’s for given masses involves solving complicated polynomial equations. Figure 2.7 shows a surface deﬁned by Euler’s quintic when one of the masses is normalized to 1. The surface lies over the mass plane in a complicated way making the uniqueness result for ﬁxed positive masses all the more remarkable. Before moving on to the proof of Moulton’s theorem we will have a look at the geometry of the next case, n = 4. This time N is the intersection of a hyperplane and an ellipsoid in R4 . So it is a two-dimensional surface diﬀeomorphic to S2 . There are six collision planes which divide the sphere into 4! = 24 triangles. Figure 2.8 shows how the collision planes divide the sphere. Proof of Moulton’s Theorem. The collision set Δ divides the ellipsoid N of normalized centered conﬁgurations into n! components, one for each ordering of the bodies along the line. Let V denote any one of these components, an open set whose boundary is contained in Δ. The Newtonian potential gives a smooth function U |V : V → R, and U (x) → ∞ as x → ∂V. Hence U |V attains its minimum at some x0 ∈ V and x0 is a CC with the given ordering of the bodies along the line. Instead of working on the normalized space where I(x) = 1 we can study the function F (x) = U (x) + I(x) on the cone V˜ of all rays through the origin passing through V (in Figure 2.6 this would be an inﬁnite triangular wedge based on one of the six arcs). Let x, y ∈ V˜ and consider a line segment p(t) = (1 − t)x + ty, 0 ≤ t ≤ 1. Note that since the ordering is ﬁxed, the sign of pi (t) − pj (t) = (1 − t)(xi − xj ) + t(yi − yj ) is equal to the common sign of xi − xj and yi − yj . It

140

Chapter 2. Central Conﬁgurations

Figure 2.7: Surface deﬁned by Euler’s quintic equations in the product space of masses and conﬁgurations. Two mass parameters (horizontal) and one conﬁguration variable r (vertical). Fixing the masses means looking for intersections of the surface with a vertical ﬁber, here a line segment. For positive masses, the segment cuts the surface just once. follows that p(t) ∈ V˜ for all t and so V˜ is a convex set. We will show that if x = y then F (p(t)) has a strictly positive second derivative. It follows that x, y cannot both be critical points of F (x). n(n−1) First consider F (rij ) as a function of the mutual distances rij on (R+ ) 2 . We have 2mi mj 2mi mj ∂2F + > 0. 2 = 3 ∂rij rij m0 Now since the conﬁgurations x, y are collinear, the mutual distances reduce to rij (t) = |pi (t) − pj (t)| and, as the ordering is constant along the segment, this is a linear function of t. It follows that F (p(t)) is a sum of terms 2 ∂2F . 2 (p(t)) rij (t) ∂rij These terms are all nonnegative and at least one is positive if x = y.

Next we will take a look at the Hessian H(x) of a collinear CC. Using the rotation invariance of U we get H(Qx) = QT H(x)Q,

2.9. Collinear central conﬁgurations

141

Figure 2.8: N for the collinear four-body problem. The collision planes divide the sphere into triangles representing the possible orderings of the bodies. where H(x) is given by (2.47) and Q ∈ SO(d) is any rotation. It follows that the index and nullity are unchanged by such rotations. If x is collinear, we can therefore assume that all of the bodies have positions xj ∈ R1 × {0}d−1 ⊂ Rd . Then the unit vectors uij appearing in the formula (2.48) are all multiples of e1 = (1, 0, . . . , 0). It follows that if we permute the components of conﬁguration vectors into groups of n with all of the e1 components ﬁrst, the e2 components next, etc., then D2 U (x) will have a block-diagonal form ˜ A, ˜ . . . , A), ˜ D2 U (x) = diag(−2A, where

⎡

A˜11

⎢m m ⎢ 13 2 ⎢ r A˜ = ⎢ 12 ⎢ .. ⎣ .

m1 mn 3 r1n

m1 m2 3 r12

···

A˜22

···

m2 mn 3 r2n

···

⎤

m1 mn 3 r1n ⎥ m2 mn ⎥ 3 ⎥ r2n

, .. ⎥ ⎥ . ⎦ A˜nn

A˜jj = −

i =j

A˜ij = −

mi i =j

3 rij

.

Note that A˜ is just the symmetric matrix A(X)M from Section 2.4. Let v = (ξ1 , ξ2 , . . . , ξd )T denote a vector in Rdn with its coordinates permuted into groups of n as described above. Vectors of the form v = (ξ1 , 0, . . . , 0)T will be called collinear vectors and those of the form v = (0, ξ2 , . . . , ξn )T normal vectors. We are interested in the tangent space Tx N to the normalized conﬁguration space. ˆ , is given by With these coordinates the center of mass subspace, ker LM m · ξi = 0,

i = 1, . . . , d,

142

Chapter 2. Central Conﬁgurations

where m ∈ Rn is the mass vector. Since x is collinear, the equation DI(x)v = 0 aﬀects only the ﬁrst vector ξ1 : m1 x11 ξ11 + · · · + mn xn1 ξ1n = 0. Finally, the action of the rotation group leads to a (d − 1)-dimensional subspace of vectors in the kernel of the Hessian. A basis is ω2 (x), . . . , ωd (x), where ωi (x) is the vector whose i-th group of n coordinates is the vector of ﬁrst coordinates of the conﬁguration, (x11 , x21 , . . . , xn1 ). For example, ω2 (x) is the tangent vector at x in the direction of a rotation in the (1, 2)-coordinate plane. Proposition 2.9.2. Every collinear central conﬁguration in Rd is nondegenerate with null(x) = d−1 and ind(x) = (d−1)(n−2). In the collinear tangent directions, H(x) is positive deﬁnite while in the normal directions it is negative semi-deﬁnite. Proof. We will analyze the Hessian block-by-block. The ﬁrst block of the Hessian corresponds to the collinear directions and we have ˜ + U (x)ξ T M ξ, ξ T H(x)ξ = −2ξ T Aξ where M is the n × n version of the mass matrix. We showed in Section 2.4 that the matrix A˜ = AM is negative semi-deﬁnite, so both terms here are nonnegative and the second is strictly positive for nonzero vectors. Therefore the collinear part of the Hessian is positive deﬁnite. For each of the other blocks we have ˜ + U (x)ξ T M ξ. ξ T H(x)ξ = ξ T Aξ The terms are of diﬀerent signs and it is a subtle problem to see which is dominant. The following proof, due to Conley, appears in [27]. Instead of ﬁnding the index and nullity of H(x) we will ﬁnd the number of negative and zero eigenvalues of the linear map with matrix M −1 H(x) = M −1 A˜ + U (X)I. It is possible to guess two eigenvalues and eigenvectors. Let u1 = 1 Since the row sums of A˜ are zero we have M −1 Hu1 = λ1 u1 ,

···

1

T

.

λ1 = U (x) > 0.

However, this vector is orthogonal to the zero center of mass subspace so is not relevant for our index and nullity computation. Next we have u2 = x = T x1 · · · xn , where we have simpliﬁed the notation so xi ∈ R denotes the position of the i-th body along the line. Then a short computation gives ˜ 2 = M −1 ∇U (x), M −1 Au

2.9. Collinear central conﬁgurations

143

where ∇ is the gradient in Rn . Since x is a normalized CC we have M −1 ∇U (x) = −U (x)x = −U (x)u2 and so M −1 Hu2 = (M −1 A˜ + U (x)I)u2 = −U (x)u2 + U (x)u2 = 0. In other words u2 is an eigenvector with eigenvalue λ2 = 0. We have one such null vector for each of the last d − 1 blocks. Note that u2 is the vector ωi (x) tangent to the rotation group action. If we can show that the other n − 2 eigenvalues of M −1 H are strictly negative, the proposition will be proved. Conley’s proof uses the dynamics of the linear ﬂow of the diﬀerential equation ˜ ξ˙ = M −1 Aξ. Every linear ﬂow determines a ﬂow on the space of lines through the origin, and the eigenvector lines are exactly the equilibrium points. Moreover the equilibrium corresponding to the largest eigenvector is an attractor for this projectivized ﬂow. If we can show that the line of the eigenvector u2 = x is an attractor, then it follows that all of the other eigenvalues of M −1 A˜ are strictly less than −U (x) and so all of the other eigenvalues of M −1 H(x) are negative. Suppose that the ordering of the bodies along the line is x1 < x2 < · · · < xn . Deﬁne a cone in the zero center of mass subspace by K = {ξ : m · ξ = 0, ξ1 ≤ ξ2 ≤ · · · ≤ ξn }. This cone contains the line spanned by the eigenvector u2 in its interior and does not contain any two-dimensional subspaces. We will show that the ﬂow carries K strictly inside itself. It follows that for the projectivized ﬂow, u2 is an attractor. Now the boundary of K is the set where one or more of the inequalities in the deﬁnition is an equality. Consider a boundary point where, for some i < j, we have ui−1 ≤ ui = · · · = uj ≤ uj+1 . The diﬀerential equation gives mk u˙ i = 3 (uk − ui ), rik

u˙ j =

k =i

mk k =j

3 rjk

(uk − uj ).

Since ui = uj the diﬀerence of these can be written: ! " 1 1 mk (uk − ui ) 3 − 3 . u˙ j − u˙ i = rjk rik k =i,j

Every term in this sum is nonnegative: if k < i, uk − ui ≤ 0 and if i < k < j,

1 3 rjk

−

1 3 rik

< 0;

1 3 rjk

−

1 3 rik

> 0.

uk − ui = 0;

if j < k, uk − ui ≥ 0 and

144

Chapter 2. Central Conﬁgurations

Moreover, not all of the terms can vanish since otherwise u would be a mul T tiple of 1 · · · 1 , which is not in the zero center of mass space. It follows that at this boundary point u˙ j − u˙ i > 0 so the point moves strictly inside the cone under the linear ﬂow. It follows that the line determined by u2 is an attractor, as required.

2.10 Morse indices of non-collinear central conﬁgurations Unfortunately, much less is known about the Morse indices of non-collinear CC’s. The following result gives a weak lower bound on the index which, at least, shows that a minimum must have the maximum possible dimension. Proposition 2.10.1. Suppose x is a central conﬁguration of the n-body problem in Rd with dim (x) < min(d, n − 1). Then the Morse index of the corresponding critical point satisﬁes ind(x) ≥ d − dim (x). In particular, the critical point is not a local minimum of U |N . As a corollary we get the existence of CC’s of the n-body problem of all possible dimensions. Corollary 2.10.2. For the n-body problem in Rd and for any k with 1 ≤ k ≤ min(d, n − 1) there exists at least one central conﬁguration with dim (x) = k. Proof. We have seen that U |N achieves a minimum at some CC x, and it follows from the proposition that dim (x) = min(d, n − 1). If 1 ≤ k < min(d, n − 1) then we can further restrict U to a subspace of Rd of dimension k and get a CC of dimension min(k, n − 1) = k. Proof of Proposition 2.10.1. If dim (x) = k < min(d, n − 1) we can assume that all of the bodies have position vectors xj ∈ W = Rk × {0}d−k . As in the last section ˜ . . . , A), ˜ we get a block decomposition of the Hessian D2 U (x) = diag(D2 (U |W ), A, 2 where D (U |W ) is the nk × nk tangential part, and where there are d − k copies ˜ We will show that the matrix M −1 A˜ + U (x)I has at of the familiar n × n block A. least one negative eigenvalue whose eigenvector has zero center of mass. Since the eigenvalue in the u1 -direction normal to the center of mass subspace is λ1 = U (x), it suﬃces to show that tr(M −1 A˜ + U (x)I) < −U (x) or, equivalently, τ = − tr M −1 A˜ > (n − 1)U (x). Now τ=

mj i

r3 j =i ij

=

mi + mj . 3 rij (i,j) i 0. To see this we will use formula (2.49). The inner product terms are 3(uij · vij )2 + 3(uij · jvij )2 = 3|vij |2

146

Chapter 2. Central Conﬁgurations

since the vectors uij and juij form an orthonormal basis for R2 . Then (2.49) gives H(x)(v, v) + H(x)(jv, jv) =

mi mj |vij |2 + 2U (q)v T M v > 0. 3 rij i 3.

2.11 Morse theory for CC’s and SBC’s In this section we will describe how to use Morse theory to prove existence of CC’s. This approach was initiated by Smale [34] and developed by Palmore [28] for the planar n-body problem, and then extended to three dimensions using equivariant Morse theory by Pacella [27]. An alternative approach to the three-dimensional case is due to Merkel [19]. Recall that central conﬁgurations in Rd , d ≥ 2, correspond to degenerate critical points of U |N due to the action of the symmetry group SO(d). In the planar case, SO(2) S1 acts freely on N \ Δ and we can think of U as a smooth function on the quotient manifold M = (N \ Δ)/ SO(2). We can still deﬁne such a quotient space when d > 2 but, due to the non-free action of SO(d), it will not be a manifold. In Section 2.8, we deﬁned the concept of nondegeneracy for CC’s with the symmetry group in mind so, using this terminology, a nondegenerate CC of the planar n-body problem determines a nondegenerate critical point in the manifold M. A generic smooth function on a manifold is a Morse function, that is, all of its critical points are nondegenerate. But it is diﬃcult to actually verify this for particular functions like the Newtonian potential. From Proposition 2.9.2 we know that the collinear CC’s are nondegenerate. When n = 3 the only non-collinear CC’s are the equilateral triangles and these are nondegenerate. The same holds for the regular simplex in the n-body problem. Proposition 2.11.1. For every choice of n positive masses, the regular simplex is a nondegenerate central conﬁguration. It is a nondegenerate minimum of the potential in the quotient space M.

2.11. Morse theory for CC’s and SBC’s

147

Proof. Suppose d = n − 1. As noted above, SO(d) acts freely on the open subset of Rn(n−1) \ Δ consisting of conﬁgurations with dim (x) = n − 1, and we can use the mutual distances rij as local coordinates in the corresponding open subset of the quotient space under rotations and translations. In these coordinates, the matrix 2 of second derivatives of F = I + U is diagonal and the partial derivatives ∂ 2 F/∂rij are all positive. Now suppose we have a curve γ(t) of normalized conﬁgurations passing through the regular simplex when t = 0, and whose tangent vector γ (0) is not in the direction of the rotational symmetry. We would like to show that U (γ(0)) > 0. The corresponding curve of mutual distances rij (t) passes through the equal-distance point corresponding to the normalized regular simplex and we have F (rij (t)) = 1 + U (γ(t)). From the discussion in the previous paragraph we have U (γ(0)) = F (rij (0)) > 0, as required. It follows that for the planar three-body problem and for all choices of the three masses, the Newtonian potential determines a Morse function on M. The space of normalized triangles is a three-dimensional ellipsoid. The quotient space under the rotation group is diﬀeomorphic to S2 and is called the shape sphere since it represents all possible shapes of triangles in the plane up to translation, rotation and scaling. M is the shape sphere with three collision shapes deleted. Figure 2.9 shows the level curves of the potential for two choices of the masses. The poles represent the equilateral triangles which are minima. On the equator, which represents the collinear shapes, there are the three collinear central conﬁgurations found by Euler, which are saddle points.

Figure 2.9: M for the planar three-body problem is the shape sphere. The Newtonian potential determines a Morse function with ﬁve critical points, shown here for the case of equal masses (left) and masses 1, 2, and 10 (right). For n > 3, d ≥ 2, it is much harder to check whether the critical points are nondegenerate. For the planar four-body problem Palmore showed that degenerate central conﬁgurations can occur for some choices of the masses and this is related to bifurcations in the number of central conﬁgurations as the masses are varied. Sim´o investigated the bifurcations numerically [33]. In Section 2.14 we will show

148

Chapter 2. Central Conﬁgurations

that for generic choices of the masses in the planar four-body problem the potential determines a Morse function. Now we will see what Morse theory tells us about the number of central conﬁgurations in the plane, taking the nondegeneracy of the critical points as an assumption. Morse theory is based on the gradient ﬂow induced by a function on a Riemannian manifold. In our case the manifold is the quotient manifold M, where we can use the restriction of the mass inner product as the Riemannian metric. First consider the gradient ﬂow on N \ Δ. If the masses are ﬁxed, Shub’s lemma allows us to restrict to a compact set of the form K = {x ∈ N : U (x) ≤ U0 } for some suﬃciently large U0 . By deﬁnition, the gradient vector ﬁeld of U |N with ˜ (x) with the respect to an inner product is the unique tangent vector ﬁeld ∇U property ˜ (x), W = DU (x)W, W ∈ Tx N . ∇U Using the mass inner product ξ, η = ξ T M η, one can check that the gradient vector ﬁeld is the restriction of ˜ (x) = M −1 ∇U (x) + U (x)x ∇U to N . By rotation invariance, this vector ﬁeld determines a gradient ﬂow on the quotient space M. Orbits of the gradient ﬂow cross the level sets of U orthogonally in the direction of increasing U . Orbits starting in the compact set K will continue to exist at least until they reach the exit level U = U0 . The Morse inequalities relate the indices of the critical points of a Morse function on a manifold M to the topology of the manifold. They are most easily expressed in terms of polynomial generating functions. Deﬁne the Morse polynomial as γ k tk , M (t) = k

where γk is the number of critical points of index k, and the Poincar´e polynomial as P (t) = β k tk , k

where βk is the k-th Betti number of the manifold, i.e., the rank of the homology group Hk (M, R) with real (or rational) coeﬃcients. Then the Morse inequalities can be written M (t) = P (t) + (1 + t)R(t), (2.52) where R(t) is some polynomial with nonnegative integer coeﬃcients. In particular, the Betti number βk is a lower bound on the number of critical points of index k. It turns out that the manifold M has a complicated topology so the Morse inequalities give interesting results. Recall that, for the n-body problem in Rd , the space N of normalized conﬁgurations is an ellipsoid of dimension d(n − 1) − 1. It is the deletion of collision set Δ which produces the topological complexity.

2.11. Morse theory for CC’s and SBC’s

149

Proposition 2.11.2. For the n-body problem in Rd , the Poincar´e polynomial of N \ Δ is P˜ (t) = (1 + td−1 )(1 + 2td−1 ) · · · (1 + (n − 1)td−1 ). In particular, for the planar three-body problem we have P˜ (t) = (1 + t)(1 + 2t) = 1 + 3t + 2t2 . Proof. It suﬃces to ﬁnd the Betti number of the unnormalized space Rdn \ Δ. To do this, note that the normalization of the center of mass and moment of inertia gives a diﬀeomorphism Rdn \ Δ Rd × R+ × (N \ Δ). Now K¨ unneth’s theorem from algebraic topology shows that the Poincar´e polynomial of a product space is the product of the Poincar´e polynomials of the factors. Here, the ﬁrst two factors are homologically trivial with Poincar´e polynomials equal to 1. The computation for Rdn \ Δ is by induction on n. For n = 1 we have Rd \ Δ = Rd \ {0} R+ × Sd−1 and we get the Poincar´e polynomial of a sphere, P˜ (t) = 1 + td−1 . For n > 1 we have a ﬁber bundle π : Rdn \ Δ → Rd(n−1) \ Δ where the projection just forgets the n-th body, π(x1 , . . . , xn ) = x1 , . . . , xn−1 ). The ﬁber over a point (x1 , . . . , xn−1 ) is Rd \ {n − 1 points} because the n-th body must avoid the other n − 1. Now this ﬁber bundle is not a product but it does satisfy certain topological conditions which guarantee that the Poincar´e polynomials multiply. First, there is a crosssection map σ : Rd(n−1) \ Δ → Rdn \ Δ with π ◦ σ = id. For example, we could let the n-th body of σ(x1 , . . . , xn−1 ) be at the point obtained by translating the barycenter of the other n−1 bodies a distance greater than the maximum distance between these bodies in the direction of the ﬁrst coordinate axis. In addition, the fundamental group of the base acts trivially on the ﬁber (for d = 2 the base is simply connected). In any case, we go from the Poincar´e polynomial for (n − 1) bodies to the polynomial for n bodies by multiplying by the Poincar´e polynomial of the ﬁber, namely 1 + (n − 1)td−1 . Next we restrict attention to the planar problem and pass to the quotient space M under the S1 action. The image of the normalized space N S2n−3 is diﬀeomorphic to the complex projective space CP(n − 2) and the projection is a nontrivial circle bundle. But when we delete the collision set, the bundle becomes trivial. For example, there is a global cross-section to the circle action consisting of all noncollision conﬁgurations where the vector from x1 to x2 is the direction of the positive ﬁrst coordinate axis. It follows that, in the planar case, N \ Δ S1 ×M.

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Chapter 2. Central Conﬁgurations

Proposition 2.11.3. For the n-body problem in R2 , the Poincar´e polynomial of the rotation reduced, normalized conﬁguration space is P (t) = (1 + 2t) · · · (1 + (n − 1)t). Proof. Since N \ Δ is the product of a circle and M, we have P˜ (t) = (1 + t)P (t). Then Proposition 2.11.2 with d = 2 gives the result. For example when n = 3, 4 we have, respectively, P (t) = (1 + 2t)(1 + 3t) = 1 + 5t + 6t2 .

P (t) = 1 + 2t,

For n = 3, the Betti numbers β0 = 1 and β1 = 2 describe the homology of the shape sphere with the three collision points deleted which is diﬀeomorphic to the twice punctured plane. To apply the Morse inequalities to the planar n-body problem ﬁrst note that we have, after quotienting by rotations, n!/2 collinear central conﬁgurations. By Proposition 2.9.2, these have Morse index n − 2. The next result, due to Palmore, uses this information to good eﬀect. Proposition 2.11.4. Suppose that all of the central conﬁgurations are nondegenerate for a certain choice of masses in the planar n-body problem. Then there are at least (3n − 4)(n − 1)! 2 central conﬁgurations, of which at least (2n − 4)(n − 1)! 2 are non-collinear. Proof. The simplest lower bound on the number of critical points is obtained by setting t = 1 in (2.52): n! γk ≥ βk = P (1) = . 2 k

k

But the information about the collinear conﬁgurations mentioned above shows that in the Morse polynomial, we have γn−2 ≥ n!/2. On the other hand, the coeﬃcient of tn−2 in the Poincar´ e polynomial P (t) is βn−2 = 2 · 3 · · · (n − 1) = (n − 1)!. Let R(t) = k rk tk be the residual polynomial in the Morse inequality (2.52). Then we have n! − (n − 1)!. rn−2 + rn−3 ≥ 2 Setting t = 1 in (2.52) now gives (3n − 4)(n − 1)! 3n! n! + 2(rn−2 + rn−3 ) ≥ − 2(n − 1)! = . γk ≥ 2 2 2 k

Subtracting n!/2 gives the non-collinear estimate.

2.12. Dziobek conﬁgurations

151

For example, when n = 3 the Morse estimate gives ﬁve critical points, which is exactly right. For n = 4 we have at least 24 CC’s of including the 12 collinear ones, assuming nondegeneracy. The estimates increase rapidly with n —we expect there to be many CC’s. In the nonplanar case, the reduction of symmetry is more complicated and the quotient space is not a manifold. See [19, 27] for two approaches to the spatial case. We also mention the paper of McCord [18] which gives estimates based on Lyusternik–Schnirelmann theory instead of Morse theory. Instead of pursuing this, we will just make a few remarks on what Morse theory can tell us about balanced conﬁgurations. Recall that these also admit a variational characterization as critical points of U |N (S) , where N (S) is the space of normalized conﬁgurations with respect to the metric based on the symmetric ˆ η. Now if we ﬁx a symmetric matrix S with distinct matrix S, ξ, η = ξ T SM eigenvalues, there is no longer any rotational symmetry and we can have nondegenerate critical points in N (S) \ Δ. The topology of this space is independent of S, so we can use the Poincar´e polynomial P˜ (t) from Proposition 2.11.2. This time there are more collinear conﬁgurations. If we ﬁx any one of the d eigenlines of S we will ﬁnd n! collinear SBC’s which are nondegenerate with Morse index (d − 1)(n − 1). There are d eigenlines for a total of d n! collinear SBC’s. If we knew their indices, it might be possible to use the information to get strong Morse estimates for the number of non-collinear SBC’s. It seems that the proof of Proposition 2.9.2 can be generalized to show that the collinear SBC’s corresponding to the largest eigenvalue of S have index (d − 1)(n − 1) which would give γ(d−1)(n−1) ≥ n!. Using this to estimate the residual polynomial as in the proof of Proposition 2.11.4 gives a lower bound γk ≥ (3n − 1)(n − 1)!, k

but this exceeds the known count of d n! collinear conﬁgurations only for d = 2.

2.12 Dziobek conﬁgurations In Section 2.9 we studied collinear central conﬁgurations. These are at the lower end of the dimension range for an n-body conﬁguration, 1 ≤ dim (x) ≤ n − 1. We also saw that the only CC with dim (x) = n − 1 is the regular simplex. In this section we consider the highest nontrivial dimension. Deﬁnition 2.12.1. A Dziobek conﬁguration is a conﬁguration of n bodies with dim (x) = n − 2. The physically interesting examples are collinear conﬁgurations of 3 bodies, planar but non-collinear conﬁgurations of 4 bodies, and spatial but nonplanar conﬁgurations of 5 bodies. They are named after Otto Dziobek who studied the planar

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Chapter 2. Central Conﬁgurations

four-body case [10]. We will be interested in ﬁnding Dziobek central conﬁgurations (DCC’s). We begin by studying the geometry of Dziobek conﬁgurations. We will assume that the dimension of the ambient space is d = n − 2 so any n-body conﬁguration is given by x = (x1 , . . . , xn ) with xj ∈ Rn−2 . It is useful to associate with x the so-called (n − 1) × n augmented conﬁguration matrix ⎡

1

⎢ ˆ =⎢ X ⎣x1

··· ···

1

⎤

⎥ ⎥. xn ⎦

(2.53)

This is just the conﬁguration matrix of Section 2.4 with a row of ones added to ˆ − 1. Note that, because of the top. Then it is easy to see that dim (x) = rank X the row of ones, two conﬁgurations are translation equivalent if and only if their augmented conﬁguration matrices have the same row space or, equivalently, the same kernel. ˆ = n − 1 and dim ker X ˆ = 1. For a Dziobek conﬁguration we have rank X Hence there is a nonzero vector Δ = (Δ1 , . . . , Δn ), unique up to a constant multiple, such that Δ1 + · · · + Δn = 0,

(2.54)

x1 Δ1 + · · · + xn Δn = 0.

ˆ k be the (n − 1) × There is a nice formula for a vector Δ satisfying (2.54). Let X ˆ k | denote (n − 1) matrix obtained from X by deleting the k-th column and let |X its determinant. Then, ˆ 1 |, −|X ˆ 2 |, . . . , (−1)k+1 |X ˆ k |, . . .)T Δ = (|X

(2.55)

is a solution to (2.54). Moreover, since the determinants are proportional to the volumes of the (n − 2)-simplices of the deleted conﬁgurations, at least one of them is nonzero in the Dziobek case. Next we will reformulate the dimension criteria above in terms of the mutual 2 . Using equations (2.54) we have distances rij or rather, their squares sij = rij j

sij Δj = |xi |2

j

Δj − 2xi ·

j

xj Δj +

j

|xj |2 Δj =

|xj |2 Δj ,

(2.56)

j

where i is any ﬁxed index and the sum over j runs from 1 to n (here, sii = 0). The result is independent of i and we denote it by −Δ0 . Deﬁne the Cayley–Menger

2.12. Dziobek conﬁgurations matrix and determinant by ⎡ 0 1 1 ⎢1 0 s 12 ⎢ ⎢1 s12 0 ⎢ CM (x) = ⎢1 s13 s23 ⎢ ⎢ .. .. .. ⎣. . . 1

s1n

s2n

153

1 s13 s23 0 .. .

··· ··· ··· ···

s3n

···

⎤ 1 s1n ⎥ ⎥ s2n ⎥ ⎥ , s3n ⎥ ⎥ .. ⎥ . ⎦

F (x) = |CM (x)|.

(2.57)

Then we have CM (x)Δ = 0, where now Δ = (Δ0 , Δ1 , . . . , Δn ). Consequently, we have F (x) = |CM (x)| = 0 for any Dziobek conﬁguration or, indeed, for any conﬁguration with dim (x) ≤ n − 2. In order to ﬁnd equations for Dziobek central conﬁgurations (DCC’s), begin by setting λ = m0 λ in the standard equations (2.10). After some algebra we ﬁnd that, for each j = 1, . . . , n, n mi Sij xi = 0, (2.58) i=1

where

1 i = j, 3 −λ , rij =− mi Sij .

Sij = mj Sjj

(2.59)

i =j

Proposition 2.12.2. Let x be a Dziobek central conﬁguration of the n-body problem, let Sij be given by (2.59) and let Δ be any nonzero solution of (2.54). Then there is a real number κ = 0 such that mi mj Sij = κΔi Δj .

(2.60)

Moreover, at least two of the Δi are nonzero. Proof. Equation (2.58) and the second equation of (2.59) show that for each j = 1, . . . , n the vector (m1 S1j , m2 S2j , . . . , mn Snj ) is a solution to equations (2.54). Since the solution is unique up to a constant multiple, there must be constants kj such that mi Sij = kj Δi . Since Sij = Sji , the vector (k1 , . . . , kn ) is a multiple κ(Δ1 /m1 , . . . , Δn /mn ) so we get (2.60) for some real number κ. If κ = 0 or if only one of the Δi were nonzero then all of the Sij , i = j, would vanish and so all of the rij would be equal. But this only happens for the regular simplex, which is not a Dziobek conﬁguration. Multiplying two of the equations (2.60) gives:

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Chapter 2. Central Conﬁgurations

Corollary 2.12.3. Let x be a Dziobek conﬁguration and let Sij be given by (2.59). Then for any four indices i, j, k, l ∈ {1, . . . , n} we have Sij Skl = Sil Skj . These equations can be used to derive some mass independent constraints on the shapes of CC’s. For example, when n = 4 we have two independent equations of the form 3 3 3 3 3 3 − λ )(r34 − λ ) = (r13 − λ )(r24 − λ ) = (r14 − λ )(r23 − λ ). (r12

Eliminating λ gives a necessary condition on the distances, in addition to the vanishing of the Cayley–Menger determinant, for a conﬁguration to be central for some choice of the masses.

2.13 Convex Dziobek central conﬁgurations In this section we present an existence proof for convex Dziobek conﬁgurations based on ideas of Xia [37]. First we discuss the geometry of the space of convex conﬁgurations. Consider the n-body problem in Rn−2 as in Section 2.12. The normalized conﬁguration space N is diﬀeomorphic to a sphere of dimension (n − 1)(n−2)−1. The Dziobek conﬁgurations form an open subset, but N also contains conﬁgurations with dim (x) < n − 2. For each x ∈ N , let Δ(x) be the vector of determinants (2.55) representing, up to a factor, the (n − 2)-dimensional volumes of its (n − 1)-body subconﬁgurations. Then Δ : N → V ⊂ Rn , where V is the hyperplane Δ1 + · · · + Δn = 0. If x is a Dziobek conﬁguration then at least two of the determinants Δi are nonzero and Δ determines a point [Δ] of the unit sphere S(V) Sn−2 in V. The planes Δi = 0 divide the sphere into components where the signs of the Δi are constant. The signs of the variables Δi provide a geometric classiﬁcation of Dziobek conﬁgurations. Suppose, for example, that Δn = 0 so that the ﬁrst n − 1 bodies span a nondegenerate simplex in Rn−2 and the ratios bi = −Δi /Δn , i = 1, . . . , n − 1, are the barycentric coordinates of xn with respect to this simplex [6]. In particular, xn is in the interior of the simplex if and only if bi > 0 for i = 1, . . . , n − 1. This provides a simple characterization of when a Dziobek conﬁguration is nonconvex, namely, we must have either exactly one Δi > 0 and Δj < 0 for j = i, or else exactly one Δi < 0 and Δj > 0 for j = i. Let N CD ⊂ N denote the open set of nonconvex Dziobek conﬁgurations. The complement K = N \ N CD is a compact set containing all of the convex Dziobek conﬁgurations. There will be some point x ∈ K where U |K achieves its minimum and we would like to conclude that x is a convex Dziobek central conﬁguration. This entails showing that the minimum does not occur on the boundary ∂K. We will prove this for n = 4 and get existence of planar, noncollinear convex central conﬁgurations for the four-body problem, a result due to

2.13. Convex Dziobek central conﬁgurations

155

MacMillan–Bartky [17]. Unfortunately, there seem to be problems extending the proof to higher dimensions. To highlight the diﬃculties, we will split the proof into two parts. First we consider the part of ∂K consisting of Dziobek conﬁgurations. This part of the proof works for all n. Proposition 2.13.1. Let x ∈ ∂K be a Dziobek conﬁguration. Then x is not the minimizer of U |K . Proof. We will show that arbitrarily close to x, there are points of K with strictly smaller values of U |K . Instead of working with normalized conﬁgurations and U |K , we can forget the normalization and use the homogeneous function G = I(x)U (x)2 . By hypothesis, there is a sequence of nonconvex Dziobek conﬁgurations xk → x. After re-indexing and taking a subsequence we may assume that for all k, the n-th body xkn is contained in the interior of the simplex formed by xk1 , . . . , xkn−1 . Taking the limit we conclude that xn is contained in the boundary of the closed simplex formed by x1 , . . . , xn−1 . Since we are assuming that x is still a Dziobek conﬁguration, x1 , . . . , xn−1 span a nondegenerate (n−2)-simplex. After re-indexing again, we may assume that xn is contained in the facet of this simplex spanned by x2 , . . . , xn−1 . Let xik , k = 1, . . . , n − 2, denote the coordinates of the bodies in the ambient space Rn−2 . After a rotation and translation we may assume x11 > 0 and xi1 = 0, i = 2, . . . , n − 1. In other words all of the bodies except x1 lie in a coordinate plane with x1 strictly to the right. Consider the distances r1k from x1 to the other bodies. Since xn is contained in the closed simplex spanned by x2 , . . . , xn−1 , we will have r1n < r1k for some k ∈ {2, . . . , n − 1} and we may assume without loss of generality that r1n < r12 . Then we will see that moving xn a little to the left while moving x2 a little to the right decreases G. Moreover these perturbed conﬁgurations are in K. We will use mutual distance version of the moment of inertia (2.9) and the usual formula for U (rij ). Note that if we move x2 , xn in the direction of the ﬁrst coordinate axis, the derivatives of the distances rij , 2 ≤ i < j ≤ n, are all zero. Only r12 and r1n change to ﬁrst order. If we change the ﬁrst coordinates of x2 , xn −1 by δx21 = m−1 2 ξ and δxn1 = −mn ξ for some small ξ > 0, a short computation shows that the ﬁrst-order change in G is −3 −3 − r1n ), δG = 2IU m1 x11 ξ(r12

where x11 > 0 is the ﬁrst coordinate of x1 . Since r1n < r12 and ξ > 0, we have δG < 0 as required. Next we need to consider boundary points x ∈ ∂K with dim (x) < n − 2. It is easy to see that every conﬁguration with dim (x) < n − 2 can be perturbed into both a convex and nonconvex Dziobek conﬁguration, hence all such lowerdimensional conﬁgurations are in ∂K. Fix a dimension k < n − 2 and let Nk ⊂ N be the set of conﬁgurations with dim (x) ≤ k. Since Nk ⊂ ∂K ⊂ K it follows that if x ∈ Nk is a minimizer of U |K then it is also a minimizer of U |Nk and is therefore a lower-dimensional CC. Therefore, in order to rule out such boundary

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Chapter 2. Central Conﬁgurations

points we need to understand how the potential changes when we perturb x to a convex Dziobek conﬁguration. We know from Proposition 2.10.1 that there will be some perturbation to a Dziobek conﬁguration which lowers the potential, but we do not know that this perturbation moves us into K. When n = 4, however, the only lower-dimensional conﬁgurations are collinear and we have the stronger Proposition 2.9.2. Proposition 2.13.2. There exists at least one convex, planar, non-collinear central conﬁguration of the four-body problem for each cyclic ordering of the bodies; hence, at least six in all, up to similarity in the plane. Proof. If x ∈ ∂K is a collinear conﬁguration, then Proposition 2.9.2 shows that every perturbation of x to a non-collinear conﬁguration in N will lower the potential. In particular, perturbing x into K will lower the potential. On the other hand, Proposition 2.13.1 shows that the non-collinear boundary points also admit potential-lowering perturbations into K. So the minimizer of U |K is in the interior as required. Note that there are six components of Dziobek conﬁgurations with Δ’s having the convex sign patterns (+, +, −, −),

(+, −, +, −),

(+, −, −, +),

and the three more with the signs reversed. These correspond to the distinct cyclic orderings. If K0 is the closure of any one of these, we can apply the same argument to ﬁnd a CC in its interior. We only need to note that the required potential-lowering perturbations can be made into K0 . In [37] it is claimed that the analogous result holds for n = 5, but as noted above, more information about the behavior of planar ﬁve-body CC’s under perturbations into Dziobek conﬁgurations seems to be needed. Given that convex Dziobek conﬁgurations exist, one can ask about their possible shapes. It is possible to use equations (2.60) together with the positivity of the masses and the signs of the Δi to derive some simple geometrical constraints, see [17, 30]. Finally, we can use the existence of at least six local minima to improve the Morse estimates for the planar four-body problem. Recall that Proposition 2.11.4 gives the existence of at least twenty four CC’s, including the twelve collinear ones (assuming that all critical points are nondegenerate). The twelve collinear CC’s have index 2 which is the maximum possible, and the six convex Dziobek conﬁgurations are minima so γ0 ≥ 6 if they are nondegenerate. The Morse inequalities become γ0 + γ1 t + γ2 t2 = 1 + 5t + 6t2 + (1 + t)(r0 + r1 t), where γ0 ≥ 6 and γ2 ≥ 12. It follows that r0 ≥ 5 and r1 ≥ 6 . Setting t = 1 gives a lower bound for the total number of CC’s of γ0 + γ1 + γ2 ≥ 12 + 2(5 + 6) = 34.

2.14. Generic ﬁniteness for Dziobek central conﬁgurations

157

This lower bound seems to be sharp although there can be as few as 32 in degenerate cases, see [12, 33].

2.14 Generic ﬁniteness for Dziobek central conﬁgurations In this section we will present a proof that there are at most ﬁnitely many similarity classes of Dziobek central conﬁgurations for generic choices of the masses; the proof is based on [22]. We will also sketch a proof that these central conﬁgurations are generically nondegenerate. Proposition 2.14.1. For generic choices of the masses, there are only ﬁnitely many Dziobek central conﬁgurations up to similarity. In fact there is a mass-independent bound on the number of such conﬁgurations valid whenever the number is ﬁnite. In particular, this applies to planar CC’s of the four-body problem and spatial but nonplanar CC’s of the ﬁve-body problem. For the four-body problem, the only non-Dziobek central conﬁgurations are the regular tetrahedron and the collinear CC’s. So in this case it follows that the total number of CC’s is generically ﬁnite. However, there is a stronger result [14]: the number of CC’s is ﬁnite for all choices of positive masses and is at most 8472. This is proved by completely diﬀerent methods which required extensive algebraic computations. Similar methods were applied to the spatial ﬁve-body problem in [13] with the result that the generic conditions on the masses mentioned in Proposition 2.14.1 are made explicit. For the planar ﬁve-body problem, Albouy and Kaloshin have recently proved generic ﬁniteness with explicit genericity conditions, see [4]. It is still open whether or not there exist exceptional choices of ﬁve positive masses which admit inﬁnitely many CC’s, but Roberts has an example involving masses of diﬀerent signs [29]. The problem of ﬁniteness for planar CC’s was singled out by Steve Smale as the sixth of eighteen problems for twenty-ﬁrst century mathematics [35]. But for n > 5 even generic ﬁniteness is open. The rest of this section is devoted to the proof of Proposition 2.14.1. The key point is to ﬁnd the dimension of the algebraic variety deﬁned by the equations for Dziobek central conﬁgurations. If the dimension of the space of central conﬁgurations is the same as the dimension of the space of normalized mass parameters, then the generic ﬁniteness will follow from general theorems of algebraic geometry. For example, in Figure 2.7, Euler’s quintic equation deﬁnes a two-dimensional surface. The projection of the surface to the two-dimensional normalized mass space necessarily has zero-dimensional ﬁbers, at least for generic masses. In this case, all of the ﬁbers are ﬁnite. We begin with equations (2.60) relating the quantities Sij from (2.59) and the Δi variables. However, we will make a few modiﬁcations. First of all, it is theoretically advantageous to work with complex, projective algebraic varieties

158

Chapter 2. Central Conﬁgurations

which are deﬁned by homogeneous polynomial equations. Deﬁne a new variable r0 such that λ = r0−3 so that −3 Sij = rij − r0−3 .

Let p = n(n − 1)/2 be the number of mutual distance variables rij . We will think of the vector r = (r0 , r12 , . . . , r34 ) ∈ Cp+1 as homogeneous coordinates for a point [r] ∈ CP(p), the complex projective space. Passing from r to [r] can be viewed as an alternative way of normalizing the size of the conﬁguration. Next we suppress the mass variables from equations (2.60) by deﬁning new variables zi = Δi /mi . After clearing denominators we get polynomial equations 3 3 r03 − rij = κzi zj r03 rij .

(2.61)

The following proposition shows that by introducing another variable z0 we can get a set of equations which are separately homogeneous in the variables r and z = (z0 , z1 , . . . , zn ) ∈ Cn+1 . We will view z as a set of homogeneous coordinates for a point [z] ∈ CP(n). Proposition 2.14.2. Suppose rij are the mutual distances of a Dziobek central conﬁguration for some choice of masses mi > 0. Let r0−3 = λ , and let [r] ∈ CP(p) be the corresponding point in the projective space. Then there is a point [z] ∈ CP(n) such that 3 3 ) = zi zj rij . (2.62) z02 (r03 − rij Moreover, the Cayley–Menger determinant vanishes, F (r) = 0. Proof. It follows from Proposition 2.12.2 and the deﬁnition of r0 that there exist zi , κ ∈ R such that (2.61) holds. Since κ = 0 we can deﬁne z0 ∈ C so that κz03 = r0−3 and then we get equations (2.62). Equations (2.62) and the Cayley–Menger determinant are separately homogeneous with respect to the variables r and z so they deﬁne a projective variety in the product space CP(p) × CP(n). As usual, we need to exclude the collision conﬁgurations. Let 0 rij = 0}. Σ = {([r], [z]) ∈ CP(p) × CP(n) : z0 r0 i 5. One could also consider the same problem in higher dimensions or for S-balanced conﬁgurations with both the masses and the symmetric matrix S ﬁxed. The generic ﬁniteness problem seems more tractable in light of Roberts’ example of a continuum of solutions for ﬁxed nonpositive masses, and the diﬃculties preventing Albouy and Kaloshin from handling all positive masses in the ﬁve-body case. Perhaps opening up the problem to allow SBC’s might make a positive mass counterexample possible. Another type of open problem is about the Morse indices of CC’s and SBC’s. As noted in Section 2.10, not much is known about the Morse indices of noncollinear CC’s and even about collinear SBC’s. Good results about this would improve the Morse theoretical estimates of the total number of critical points. It was a lack of information about the Hessian in directions normal to the subspace occupied by the conﬁgurations which prevented us from extending the existence proof for convex Dziobek conﬁgurations to n > 4 bodies. The most natural conjecture, that the normal blocks of the Hessian are negative semi-deﬁnite, is not true in general. There are planar CC’s for which the potential increases in certain normal directions [20, 24].

2.15. Some open problems

163

As far as we know, the convex Dziobek conﬁgurations of the four-body problem are unique given the ordering of the bodies, but no proof has been given. The problem of counting convex Dziobek conﬁgurations could be posed for n > 4 once the existence problem is solved. Another group of open questions concerns a topic not treated in these notes, namely the dynamical stability of relative equilibrium and homographic motions. Given a planar CC we saw that we have a simple relative equilibrium solution where the bodies rigidly rotate around their center of mass. In rotating coordinates this becomes an equilibrium and one can ask about its linear stability. In particular, one can ask if there is any relation between the eigenvalues at the equilibrium point and the Morse index of the critical point. All of the known examples of linearly stable relative equilibria correspond to critical points which are local minima. Is this always the case? In light of Albouy–Chenciner’s theory of higher-dimensional relative equilibria, one can generalize the problem to ask for the relationship between the properties of an SBC as a critical point and as an equilibrium point of the reduced equations of motion. In fact, the problem of linear stability of higher-dimensional relative equilibria seems to be completely open.

Bibliography [1] A. Albouy, Recherches sur le probl´eme des conﬁgurations centrales, preprint (1997). [2] A. Albouy, Mutual distance in celestial mechanics, Lecture Notes from Nankai University, Tianjin, China, preprint (2004). [3] A. Albouy and A. Chenciner, Le probl`eme des n corps et les distances mutuelles, Inv. Math. 131, (1998) 151–184. [4] A. Albouy and V. Kaloshin, Finiteness of central conﬁgurations of ﬁve bodies in the plane, Annals of Math. 176, (2012) 535–588. [5] V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., SpringerVerlag, New York, (1989). [6] M. Berger, Geometry I, Springer-Verlag. [7] A. Chenciner, The Lagrange reduction of the N -body problem, a survey, arXiv:1111.1334, (2011). [8] D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, Springer-Verlag, New York, (1997). [9] H.S.M. Coxeter, Regular Polytopes, Dover, New York, (1973). ¨ [10] O. Dziobek, Uber einen merkw¨ urdigen Fall des Vielk¨ orperproblems, Astron. Nach. 152, (1900) 33–46. [11] L. Euler, De motu rectilineo trium corporum se mutuo attrahentium, Novi Comm. Acad. Sci. Imp. Petrop. 11, (1767) 144–151. [12] M. Hampton, Concave Central Conﬁgurations in the Four Body Problem, thesis, University of Washington (2002). [13] M. Hampton and A. Jensen, Finiteness of spatial central conﬁgurations in the ﬁve-body problem, Cel. Mech. Dyn. Astr. 109, (2011) 321–332. [14] M. Hampton and R. Moeckel, Finiteness of relative equilibria of the four-body problem, Inventiones Mathematicae 163, (2006) 289–312.

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[15] J.L. Lagrange, Essai sur le probl`eme des trois corps, Œuvres 6, (1772). [16] R.C. Lyndon, Groups and Geometry, London Mathematical Society Lecture Notes Series 101, Cambridge University Press (1985). [17] W.D. MacMillan and W. Bartky, Permanent Conﬁgurations in the Problem of Four Bodies, Trans. Amer. Math. Soc. 34, (1932) 838–875. [18] C.K. McCord, Planar central conﬁguration estimates in the n-body problem, Ergod. Th. and Dynam. Sys. 16, (1996) 1059–1070. [19] J.C. Merkel, Morse theory and central conﬁguration in the spatial N -body problem, J. Dyn. Diﬀ. Eq. 20, (2008) 653–668. [20] R. Moeckel, On central conﬁgurations, Math. Zeit. 205, (1990) 499–517. [21] R. Moeckel, Relative equilibria with clusters of small masses, Jour. Dyn. Diﬀ. Eq. 9, (1997). [22] R. Moeckel, Generic Finiteness for Dziobek Conﬁgurations, Trans. Amer. Math. Soc. 353, (2001) 4673–4686. [23] R. Moeckel, Celestial Mechanics – especially central conﬁgurations, available at http://www.math.umn.edu/∼rmoeckel/notes/Notes.html. [24] R. Moeckel and C. Sim´o, Bifurcations of spatial central conﬁgurations from planar ones, SIAM J. Math. Anal. 26, (1995) 978–998. [25] F.R. Moulton, The Straight Line Solutions of the Problem of n Bodies, Ann. of Math. 12, (1910) 1–17. [26] D. Mumford, Algebraic Geometry I, Complex Projective Varieties, Grundlehren der Mathematische Wissenschaften 221, Springer-Verlag, Berlin, Heidelberg, New York (1976). [27] F. Pacella, Central conﬁgurations of the n-body problem via equivariant Morse theory, Arch. Rat. Mech. 97, (1987) 59–74. [28] J. Palmore, Classifying relative equilibria, I: Bull. AMS, 79 (1973) 904-908; II: Bull. AMS, 81 (1975) 489–491; III: Lett. Math. Phys. 1, (1975) 71–73. [29] G. Roberts, A continuum of relative equilibria in the ﬁve-body problem, Phys. D 127, (1999) 141–145. [30] D. Schmidt, Central conﬁgurations in R2 and R3 , in Hamiltonian Dynamical Systems, Contemporary Math. 81, (1988) 59–76. [31] I.R. Shafarevich, Basic Algebraic Geometry 1, Varieties in Projective Space, Springer-Verlag, Berlin, Heidelberg, New York (1994). [32] M. Shub, Appendix to Smale’s paper: Diagonals and relative equilibria, Lecture Notes in Math. 197, (1971) 199–201.

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[33] C. Sim´o, Relative equilibria in the four-body problem, Cel. Mech. 18, (1978) 165–184. [34] S. Smale, Problems on the nature of relative equilibria in celestial mechanics, Lecture Notes in Math. 197, (1971) 194–198. [35] S. Smale, Mathematical problems for the next century, Mathematical Intelligencer 20, (1998) 7–15. [36] Z. Xia, Central conﬁgurations with many small masses, J. Diﬀerential Equations 91, (1991) 168–179. [37] Z. Xia, Convex central conﬁgurations for the n-body problem, J. Diﬀerential Equations 200, (2004) 185–190.

Chapter 3

Dynamical Properties of Hamiltonian Systems with Applications to Celestial Mechanics Carles Sim´ o 3.1 Introduction Our goal is to study some properties of the dynamics of the N -body problem. As is well known, the Newtonian model of N punctual masses, mi , i = 1, . . . , N , located at qi (t) ∈ Rd , moving under their mutual gravitational attraction is described by the equations q¨i =

N

3 (qj − qi )/ri,j ,

2 ri,j = ||qj − qi ||22 ,

i = 1, . . . , N.

(3.1)

j=1,j =i

The system has several ﬁrst integrals. The centre of mass ones, in a suitable N N reference moving linearly with constant velocity, are i=1 mi qi = 0, i=1 pi = 0, where the related momenta are deﬁned as pi = mi q˙i . Furthermore, deﬁning N 2 the kinetic energy as T (p) = i=1 ||pi ||2 /mi and the potential one as U (q) = 1≤i 0. Note that, eventually, some Q can never return to Σ for any t > 0. This implies that Σ has to be reduced to a suitable subset. We also note that the return time t(Q) depends on the starting point. We denote as P(Q) := ϕt(Q) (Q) the image of Q under the Poincar´e map P. In the case of a Hamiltonian H with m d.o.f. (hence x has dimension 2m) ﬁxing a transversal section Σ and the level of energy h, the Poincar´e map associated

3.1. Introduction

171

to Σ deﬁnes a map in Σ ∩ H −1 (h), of even dimension 2(m − 1). This map is symplectic. Given a discrete map x → F (x) in V ⊂ Rn , there is also a simple way to produce a v.f. such that it has, as associated Poincar´e map, the initial map, provided F is close to the identity, say F (x) = x + εG(x) with ε small enough (see later). For concreteness we shall assume that G is a real analytic function. We want to deﬁne a non-autonomous periodic v.f. of period 1 in t. Let us consider, t for instance, and for a given k > 1, the function ψk (t) = c 0 sk (1 − s)k ds, where the constant c is selected to have ψk (1) = 1. Then we deﬁne the ﬂow starting at the point x after a time t ∈ [0, 1] as ϕt (x) = x + εψk (t)G(x), that is, we are using an Hermite-like interpolation, because ψkj (0) = 0 for j = 0, . . . , k, ψk (1) = 1, and ψkj (1) = 0 for j = 1, . . . , k. Other interpolations can also be used. For other values of t it is deﬁned by periodicity: ϕt (x) = ϕ(t) (x) where (t) = t − [t], being [t] the largest integer less than or equal to t. Clearly ϕ0 (x) = x, ϕ1 (x) = F (x). Now we should deﬁne the v.f. at (y, τ ) for τ ∈ [0, 1]. To this end we look for z such that ϕτ (z) = y. It follows immediately, from the implicit function theorem, that k a solution exists if || Id + εDG||∞ > 0. Finally the v.f. is f (y, τ ) = ε dψ dt (τ )G(z). We note that this is a slow v.f., having the parameter ε as a factor. It is usually referred to as the suspension of the map F . We can consider if it is possible to approximate it by an autonomous v.f. This follows from a general theorem on averaging, that we present in a wider context: the case of v.f. depending on time in a quasiperiodic way. Theorem 3.1.1. Let z˙ = εf (z, θ, ε),

(3.2)

where f is analytic in (z, θ) for z ∈ Ω ⊂ Cn , Ω = D + Δ, a Δ-neighbourhood of D in Cn , D a compact in Rn , and θ ∈ Tp + Δ, p ≥ 2, where Tp is a p-dimensional torus. Assume f in (3.2) is bounded in ε for |ε| ≤ ε0 and θ = ωt, where ω ∈ Rn is a vector of frequencies satisfying the Diophantine condition (DC) |(k, ω)| ≥ b|k|−τ , ∀k ∈ Zp \ {0}, (3.3) p where b > 0, τ > p−1 and |k| = i=1 |ki |. Then, if ε0 is small enough, for a ﬁxed ε with |ε| ≤ ε0 , there exists a change of variables z = h(w, θ, ε), analytic in (w, θ) for w ∈ D+Δ/2, θ ∈ Tp +Δ/2, such that the new equation is w˙ = ε(g(w, ε)+r(w, θ, ε)) and the remainder satisﬁes an exponentially small bound |r|Δ/2 < c1 exp(−c2 /εc3 ),

(3.4)

where c1 , c2 > 0 , c3 = 1/(τ + 1). The constants c1 , c2 depend only on |f |Δ , the dimensions and the constants in (3.3). Furthermore, |g|Δ/2 < 2|f |Δ . Here |f |Δ denotes the sup norm of f in D + Δ, Tp + Δ for the ﬁxed value of ε.

172

Chapter 3. Dynamical Properties of Hamiltonian Systems

Remark 3.1.2. (i) In the periodic case (it would be p = 1, τ = 0), there is no need of analyticity with respect to t; just integrable is enough. Then c3 = 1, see [36]. (ii) The optimal number of averaging steps (i.e., up to which order in ε one should cancel the quasiperiodic dependence) is ≈ ε−c3 . (iii) If f is a Hamiltonian v.f., the change to w can be made canonical. Hence, the averaged system, skipping the remainder r, is also Hamiltonian, see [51]. (iv) If f has been obtained by suspension of a map F , we can produce an autonomous v.f., like g, which interpolates F except by exponentially small terms. The basic idea of the proof is to obtain the change z = h(w, θ, ε) by means of sequence of changes. This methodology is common to many topics in dynamical systems. First we try to cancel the purely quasiperiodic terms in f , that is, the terms in f˜ = f − f¯, where f¯ denotes the average with respect to θ. Writing the suitable condition for the change, one has to solve a PDE to obtain the quasiperiodic coeﬃcients in this ﬁrst change. To solve it with control on how the coeﬃcients of the change behave is where the analyticity with respect to θ and the DC (3.3) play their roles. In the periodic case one has to do just an integration, and this is why to be integrable in t is enough in that case. Once the terms in f˜ have been skipped, one has to check the contribution that the change makes in ε2 . Here is where the analyticity with respect to z plays a role, to bound the derivatives in a slightly smaller strip, passing from half width Δ to Δ1 . Then we proceed to cancel the purely quasiperiodic terms which appear with ε2 as factor, and so on, to cancel the non-autonomous terms in εk , k = 3, . . . At every step, to be able to bound the contributions made by the change to higher order in ε, one has to reduce the size of the analyticity domain, introducing a decreasing sequence for the half widths of the successive domains Δ2 > · · · > Δk > · · · . After every change one has a bound on the remainder. If for a given ε we do too many changes, as we want to keep an analyticity domain of positive half width, the diﬀerences Δk−1 − Δk are small. This implies bad estimates for the derivatives and an increase on the size of the remainder. This is why, for every ε, there is an optimal order. Simpler estimates give then the bound in (3.4). See [41] for details and examples. These kind of bounds on remainders are relevant to bound errors on approximations done, for instance, with normal forms (see Subsection 3.3.2). The variables can be scaled in the domain of interest and the role of ε is played by the size of the domain. Finally we stress that the passage from ﬂows to maps and vice-versa, when the map or some power of it is close enough to the identity, allows a more complete understanding and representation of key phenomena.

3.2. Low dimension

173

In what follows we shall consider that all v.f. and maps are in the analytic category.

3.1.2 Comments on the contents Setting aside the two-body problem and subclasses with some special symmetry, the simplest N -body problem is the planar circular restricted three-body problem which has two d.o.f. (see Section 3.4). The next simplest problem can be the planar general three-body problem. Even restricting to a ﬁxed value of the angular momentum it has three d.o.f. The dimension can be reduced by ﬁxing energy and using a Poincar´e section. In the ﬁrst case we obtain symplectic 2D maps, easy to visualize. In the second case we have symplectic 4D maps, not so easy to visualize. There are key objects of codimension one (see Subsection 3.3.4) and homoclinic/heteroclinic phenomena due to the intersection of two objects of dimension two in dimension four. The invariant tori (see Subsection 3.3.3) do not separate the phase space, and slow escape from points as close as we like to invariant tori (Arnold diﬀusion or general diﬀusion, see Subsection 3.3.5), avoiding a large set of nearby tori, can occur. For these reasons we devote Section 3.2 to introducing several simple but paradigmatic examples in the 2D case, with the hope that they will make it easier to grasp the main ideas in higher dimension. See also slides (B) for several examples with low dimensional conservative systems. Section 3.3 is devoted to presenting some general theoretical results. But it is also relevant to see how to use the ideas of the proofs in concrete examples. In many cases, eﬀective computation is based on implementation of the proof, either by symbolical or numerical methods or, quite frequently, by a combination of both. Finally Section 3.4 presents some applications to Celestial Mechanics, with a variety of goals. Concerning references, most of the basic results can be found in classical standard books. A few of them appear in the list of references, and no explicit mention to them is made in the text. Some references to concrete topics are scattered throughout the text and they are given at the end of these notes. The reader can also, at the end of the references, look at the list of slides of several talks given in the past and that, in turn, refer to some animations.

3.2 Low dimension: same key examples of 2D symplectic maps to see the kind of phenomena to face Invariance of dx ∧ dy in dimension 2 is equivalent to area preservation. We shall denote as APM the area preserving maps. The simplest non-trivial APM which come to mind are the quadratic ones: x, y ∈ R and F1 , F2 are polynomials of degree

174

Chapter 3. Dynamical Properties of Hamiltonian Systems

two. These maps were widely studied by M. H´enon [20]. See also slides (E) and slides (G). They are relevant because: (i) the number of parameters can be reduced to only one, and they have a very simple geometrical interpretation; (ii) they appear in a natural way as a very good approximation in some parts of R2 when we consider arbitrary APM; in particular when we study Poincar´e maps of Hamiltonian systems with two d.o.f.; (iii) the following problems can all be understood thanks to our knowledge of the quadratic case: (a) the existence of invariant curves diﬀeomorphic to S1 ; (b) the role of the invariant manifolds of hyperbolic ﬁxed or periodic points and how they lead to the existence of chaos; and (c) the geometrical mechanisms leading to the destruction of invariant curves. We shall illustrate some of these features in this section.

3.2.1 The H´enon map In the initial formulation the map (except in some degenerate cases) can be written, thanks to the APM character, shift of origin and scaling, as F : (x, y) → (1 − ax2 + y, −x). Hence, this family of maps depends on a single parameter a. The geometric interpretation is simple: it is the composition of the map (x, y) → (x, y + 1 − ax2 ) (one of the so-called de Jonqui`eres maps) and a rotation of angle −π/2. Figure 3.1 shows, for a = −1/2, the square [−3, 3]2 (in red), the ﬁrst image (in green) and part of the next two images (in blue and magenta, respectively). One can ask whether all points will escape for future iterations. To give an answer to this question, we plot in black the set of points which remain bounded for all iterations and the selected value of a. However, we shall use another representation of that map, see [52], given by

Fc

x y

→

x + 2y + 2c (1 − (x + y)2 ) y + 2c (1 − (x + y)2 )

,

(3.5)

which depends on c that can be assumed to be positive. It has two ﬁxed points: H at (−1, 0), hyperbolic ∀c > 0, and E = (1, 0), elliptic for 0 < c < 2 and hyperbolic with reﬂection for c > 2. The reversor S(x, y) = (x, −y) allows us to obtain Fc−1 = SFc S. √ √ Doing the change of scale (ξ, η) = (x, 2y/ c) one obtains a map c-close to √ the identity. According to Section 3.1 it can be approximated by the time- c ﬂow of the v.f. dξ/dt = η, dη/dt = 1 − ξ 2 , with Hamiltonian K(ξ, η) =√η 2 /2 − ξ + ξ 3 /3. It is, of course, a trivial matter to improve K to any power of c. This v.f. has the same ﬁxed points as Fc and a separatrix on the level K = 2/3.

3.2. Low dimension

175

Next we show iterates of some initial points under Fc for c = 0.2 and c = 0.762, see Figure 3.2. 3 2 1 0 -1 -2 -3 0

-2

4

2

6

8

2

Figure 3.1: The square [−3, 3] (in red) and the ﬁrst three images of it under the H´enon map with a = −0.5, shown in green, blue and magenta, respectively. The last two have parts outside the frame shown here. In black we display the invariant set of points which remain bounded under all iterations.

0.4

0.6

0.2

0.3

-0.2

-0.3

-0.4

-0.6 -1

1

2

0.5

1

1.5

Figure 3.2: Some iterates under Fc . Left: for c = 0.2. Right: for c = 0.762. We have taken initial points on y = 0 and plotted 5,000 iterates of each one after a transient of 106 iterates. Points outside the displayed domain escape to inﬁnity close to the left u . When looking at the ﬁgure in the electronic version it is suggested to branch of WH magnify the right plot to see the details. The same applies to several other plots in the next ﬁgures, without explicit mention. An important characteristic of points whose orbit is an invariant curve (IC) is the rotation number ρ. It measures the average value of the fraction of revolution that the point turns in each iterate. We can look at the curves around the elliptic point E in previous plots and take polar coordinates. Let θk be the angle of the k-th iterate, but considered in the lift R instead of S. Note that in this example

176

Chapter 3. Dynamical Properties of Hamiltonian Systems

the points turn clockwise. Then, we deﬁne ρ=

θk 1 lim . 2π k→∞ k

(3.6)

It always exists and does not depend on the initial point on the curve. On the left plot in Figure 3.2 one can see a pattern which looks like the phase portrait of a one d.o.f. Hamiltonian, with a foliation by periodic solutions and a separatrix in blue. It seems that, as in the case of one d.o.f. systems, the map is integrable. That is, there exists a non-constant function C(x, y) preserved by the map: C(F (x, y)) = C(x, y). In fact there is a Cantor set (of positive measure) of ICs with ρ ∈ / Q, an inﬁnite number of periodic orbits of elliptic and hyperbolic type and the right-hand sides of the manifolds of H do not coincide. What happens is that the diﬀerences with respect to the ﬂow case are extremely small, in agreement with (3.4). We shall see details on this smallness in the part about invariant manifolds of Section 3.2.1. The right-hand plot in Figure 3.2 displays a typical behaviour of a not close to integrable APM. Certainly there are many ICs (again a Cantor set) around the point E, but at some distance one can see big period-5 islands around elliptic periodic points of period 5, and one can guess the existence of period-5 hyperbolic points. Close to them there are chaotic orbits, still surrounded by some more ICs, (rotational, that is, they make the full turn around E) and, ﬁnally, some little islands before reaching a place where most of the points escape. Some comments on invariant curves The plots in Figure 3.2 raise several questions: (1) do ICs really exist for Fc ? (2) what is the structure of the set of ICs? (3) how are they destroyed? (4) what happens after an IC’s destruction? First we introduce the so-called twist maps. These are integrable maps deﬁned in some annular domain rd < r < ru , having a foliation by ICs, given by T (r, α) = (r, α + a(r)),

(3.7)

and satisfying the twist condition da(r)/dr = 0.

(3.8)

Of course, one can have the form (3.7) after a diﬀeomorphism. The curves can have a shape diﬀerent from circles, like ellipses, to be star-shaped or not. A key result is the Moser twist theorem. Theorem 3.2.1. Consider a perturbation Fε = T + εP of a twist map T . Then, if we have an invariant curve of T which has Diophantine rotation number γ, this curve, with a small deformation, subsists for Fε provided ε is suﬃciently small.

3.2. Low dimension

177

The Diophantine condition, in the present case, is like (3.3) with frequencies γ and 1: |k1 γ + k0 | ≥ b|k|−τ , ∀(k1 , k0 ) ∈ Z2 \ {0}, where |k| denotes some norm of k = (k1 , k0 ). Let us comment a little on the three conditions: (a) it must be a perturbation of a twist map T ; (b) the rotation number γ must be Diophantine; and (c) it must be close enough to T , that is, ε must be small. Assume that the Fourier representation of the IC of T which has ρ = γ is r(α) = j∈Z aj exp(i jα) in the present polar coordinates we are using (typically, for a given problem, the twist map will not be given in the form (3.7), and to put it explicitly in this form can be cumbersome). Let rε (α) = j∈Z bj exp(i jα) be the representation of the desired IC, invariant under Fε . The invariance condition is expressed, in (r, α), as Fε (rε (α), α) = (rε (α + 2πγ), α + 2πγ). It is clear that we can ﬁx the origin of angles in an arbitrary way. We try to pass from the coeﬃcients aj to bj by making a sequence of changes (similar to the case of Theorem 3.1.1) such that, after the k-th change, one has k an approximation of the IC under Fε with ρ = γ with an error O(ε2 ). That is, a Newton method in the space of Fourier series. The equation to be solved at each step is of the form G(α + 2πγ) − G(α) = R(α), the so-called homological equation, where R(α) is related to the error of the previous approximation and has zero average, a necessary condition in order to make it possible to solve the equation. Using Fourier representations for G and R, say G = j∈Z gj exp(i jα) and R = j∈Z rj exp(i jα), r0 = 0, it is straightforward to obtain gj = rj /(exp(i j2πγ) − 1), j = 0. But, of course, if jγ is close to an integer, the previous denominator is close to zero. This is known as the small denominators problem. The DC allows us to control the behaviour of the coeﬃcients of G, so that if R is analytic in some complex strip around real values of α, G is also analytic (perhaps in a slightly narrower strip). The problem is then that the error in the next approximation does not have zero average and we will not be able to solve the next homological equation. But this average can be canceled by modifying the initial independent term a0 (or, equivalently, by selecting a proper value for g0 ) and this is possible, thanks to the twist condition, by applying the implicit function theorem. It is convenient to express the twist condition as dρ/da0 = 0; that is, in terms of the average of the initial curve. Finally the smallness of ε is necessary to have convergence in the Newton procedure. Note that, for a ﬁxed γ, the larger is the twist condition, the larger are the admissible values of ε. In Section 3.3.3 we shall talk about generalizations to higher dimension, both for symplectic maps and for Hamiltonian ﬂows. The key ideas for the proofs are the same. Using normal form tools (see Section 3.3.2) it is easy to prove that, what seem to be ICs in Figure 3.2 are really ICs, at least close to the point E. Furthermore, it is clear that the structure of the set of ICs is Cantorian, because so is the structure of the set of Diophantine numbers for values of b, τ bounded from below, see (3.3).

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It is interesting to see what happens when the twist condition is not satisﬁed. Figure 3.3 shows, for the map (3.5), with c = 1.35, the evolution of ρ as a function of x for initial points of the form (x, 0). It is clear that ρ is only deﬁned for ICs and periodic orbits (or islands) and, in the present case, it seems that this occurs for most of the initial values of x. One can prove that this behaviour, with a local minimum at x = 1 and a local maximum on each side, appears only for c ∈ (c1 , c2 ), c1 = 5/4, c2 ≈ 1.4123. For c ∈ (0, c1 ) one has a local (in fact, global) maximum at x = 1 (the point E).

0.308

0.306

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0.302 0.4

0.6

0.8

1

1.2

Figure 3.3: For c = 1.35 the value of ρ = ρ(x) is plotted for initial points on y = 0.

In blue the points with ρ ∈ Q. Note that now, to the left or to the right of x = 1, the function ρ is no longer monotonous.

The twist condition is lost near the maxima. Let ρM be the value of ρ at a given maximum M . Assume that there exist rationals p/q < ρM with q not too large. They give rise to the typical islands structure, with q islands on each family, on both sides of the IC with ρ = ρM (or close to it). The interaction of these two families of islands, with ρ = p/q, gives rise to the so-called meandering curves, see [43], which cannot be written with the radius as a function of the angle seen from the point E. The curves have some folds (or meanders) but it is still possible to apply Moser’s Theorem 3.2.1 to prove that they exist. Figure 3.4 shows an example of meandering ICs for c = 1.3499 and a magniﬁcation including some nearby orbits.

Some comments on invariant manifolds of hyperbolic points Beyond the IC of an APM, there are other very important invariant objects which play a key role in dynamics (this is also true for more general maps and ﬂows in any dimension, see Section 3.3.4). They are the stable and unstable manifolds of the hyperbolic ﬁxed or periodic points. They can be seen as the non-linear generalisation of the invariant subspaces of the diﬀerential of the map at the ﬁxed point. On the left-hand plot in Figure 3.2, for c = 0.2, the branches W u,+ and W s,− (the ones which start to the right of x = −1) seem to be coincident, but they are not.

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0.3 0.04 0.2 0.02

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-0.1

-0.02

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0.9

0.8

1.1

0.76

0.77

0.78

0.79

Figure 3.4: Left: We show a couple of orbits for c = 1.3499, sitting on a domain in which

ρ passes through a maximum. These orbits are on invariant curves known as meanders. Right: A magniﬁcation of the left. Beyond diﬀerent meanders in red, one can see two typical invariant curves (inner and outer) in blue and islands, in magenta, which belong to two diﬀerent chains of islands of rotation number 4/13.

Figure 3.5, left, shows a magniﬁcation when they return to the vicinity of the point (−1, 0), after going clockwise around E under Fc (red points), or counterclockwise under Fc−1 (blue points). We see tiny oscillations with a size O(10−3 ). The righthand plot in Figure 3.5 shows the manifolds for c = 0.77 with large oscillations. The points in W u ∩ W s are known as homoclinic points (or biasymptotic points). Some of them, on y = 0, can be seen to the right of the plot. The successive nearby returns of the manifolds produce inﬁnitely many homoclinic points. Depending on the location of a point with respect to a given homoclinic, after passing close to H under iteration by Fc , will follow close to the positive, W u,+ , or to the negative, W u,− , branches of W u . And similarly for the stable branches using Fc−1 . 0.004 0.8 0.002

0.4

0 -0.4

-0.002

-0.8 -0.004 -1.01

-1.005

-1

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1

2

Figure 3.5: Left: a magniﬁcation of Figure 3.2 showing that the manifolds do not coincide. Right: part of the invariant manifolds of the hyperbolic point H for c = 0.77 (the s u unstable manifold in red, the stable one in blue). One has WH = S(WH ). The splitting of the manifolds is now clearly visible. It is increasing with c. Note that the domain around the point E which is not covered by the oscillations of the manifolds, becomes smaller. Compare with the non escaping set of points in Figure 3.2 right, for a nearby value of c.

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A measure of the lack of coincidence of W u and W s is the splitting angle. This is deﬁned as the angle between manifolds at a given homoclinic point. In the present case of quadratic APM, we can measure the angle at the ﬁrst intersection of the manifolds with y = 0 to the right of x = 1 and see how it behaves as a function of c. For concreteness, we denote this angle as σ(c). In Figure 3.6, left and middle, we represent the value of σ(c) in diﬀerent scales. In the left-hand plot, despite the splitting being diﬀerent from zero for any c > 0, we see that only for c > 0.2 does it start to be visible. To make visible what happens for small c, we display, in the middle plot, log(σ) against log(c). Note that already for c = 0.05 the value of σ(c) is below 10−15 and, hence, it is negligible for any practical application. 0.08

0.06

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-8

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50

100

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Figure 3.6: Diﬀerent representations of the splitting angle σ(c) between the mani-

folds at the ﬁrst intersection with y = 0, x > 1. Left: σ, as a function of c, showing that σ seems negligible for c < 0.2. At that value of c, the ﬁrst digits are σ(c) = 6.2146342685682663009767540674985307425003 . . . × 10−5 . Middle: log(σ) as a function of log(c), which allows to see how small σ(c) is for c approaching zero. Right: the values of log10 (ω2m (2π 2 )2m /(2m + 6)!) versus m, to give evidence of the Gevrey character of Ω(h) (see text).

Concerning the right-hand plot in Figure 3.6 we need some preliminaries. Let √ λ(c) be the dominant eigenvalue at the point H, which is equal to 1 + c + 2c + c2 for Fc . An essential parameter in the theoretical study of the problem is h(c) = log(λ(c)), because using suitable representations of the manifolds, it is possible to show that the splitting has upper bounds of the form exp(−η/h), where η is related to the imaginary part of the singularity of the separatrix of the limit ﬂow, as mentioned before in Figure 3.2. This type of result is true for general analytic APM close to the identity map, see [10, 11]. In fact, for the present problem one can prove a more precise result. The splitting angle has the form

9 2π 2 σ(c) = × 106 π 2 h(c)−8 exp − × Ω(h). (3.9) 2 h(c) The term Ω(h) can be expanded in powers of h2 , say Ω(h) = m≥0 ω2m h2m , and can be bounded by ω0 + O(h). The constant term can be determined numerically and the ﬁrst digits of ω0 are 2.48931280293671. Note, however, that the series deﬁning Ω(h) is divergent. But for every value of h it provides a good approxima-

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tion if we truncate the summation at the appropriate place. There is numerical evidence that the series is of Gevrey-1 class. class if the series A formal power series ck tk is said to be of Gevrey-β −β k ck (k!) t is convergent. We can compute the series m≥0 ωm h2m /(2m)! obtained from Ω(h) using β = 1. From a numerical determination of Ω(h), for diﬀerent values of h, one can obtain the coeﬃcients ω2m . See [12] for methodology and examples. In the right-hand plot in Figure 3.6 we display log10 (ω2m (2π 2 )2m /(2m+ 6)!) as a function of m, which seems to tend to a constant. This gives evidence of the Gevrey-1 character of Ω(h) we mentioned. But to prove this fact is an open problem. See slides (H) for the role of the splitting phenomena in the measure of the chaotic domain in diﬀerent problems. On the destruction of invariant curves As mentioned in the part about invariant submanifolds in Section 3.2.1, if ρ is too close to a rational (in the Diophantine sense), or if the twist condition is too weak, or if the perturbation ε with respect to an integrable map is too large, the IC does not exist. These analytic properties also have a nice geometric interpretation. To illustrate the mechanism leading to the destruction of ICs we consider Figure 3.7. It has been produced for c = 0.618 (left) and c = 0.63 (right) and it only shows the left-hand part of the set of points which have bounded orbits. The case of Figure 3.7 is similar to the one on the Figure 3.2 right, but now the main islands are 6-periodic instead of 5-periodic. On the left-hand plot one can see medium size islands with ρ = 3/19 (one of them with its central elliptic point on y = 0) and two symmetrical islands, in the same family, with ρ = 4/25, as well as several satellite islands, then tiny islands (e.g., with ρ = 17/107, 39/245, 11/69, 19/119, 21/131, 13/81, . . .) and ICs. In particular, some ICs are still present between the two chains of medium size islands. Some other ICs, surrounding the main period-6 islands (not displayed), appear as the rightmost curves shown. On the right-hand plot we display in black two chains of islands of rotation numbers 3/19 and 4/25, corresponding to the ones in the left plot, but now they are smaller. Consider the associated hyperbolic periodic orbits, the one with rotation number 4/25 being visible on the x-axis and the two symmetric points belonging to the hyperbolic periodic orbit of rotation number 3/19 being close to x = −0.2 oﬀ the x-axis. The manifolds of these periodic orbits give rise to heteroclinic connections, that is, intersections of the stable and unstable maniu s folds of two diﬀerent objects. The manifolds W4/25 , W4/25 are shown in red and u s green, respectively. The manifolds W3/19 , W3/19 are shown in blue and magenta, u s s u respectively. Note that W4/25 and W3/19 (and, symmetrically, W4/25 and W3/19 ) have transversal heteroclinic intersections. This produces an obstruction to the existence of the ICs which could separate the chains of islands. This is the basis of the so-called obstruction mechanism [38].

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Figure 3.7: Left: A part of the set of non escaping points for the map Fc and c = 0.618.

Right: similar plot for c = 0.63, displaying also several invariant manifolds of periodic hyperbolic points leading to heteroclinic intersections. See the text for details.

Indeed, if we consider a curve formed by a piece of invariant manifold of the inner hyperbolic periodic point (the one of period 25) until the heteroclinic point, followed by a piece of invariant manifold of the outer hyperbolic periodic point (the one of period 19), from the heteroclinic point to the periodic one, the ICs will have to cross it. This is impossible because of the invariance. In fact, one concludes that ICs with ρ ∈ (3/19, 4/25) cannot exist. But ICs with ρ in that interval are found for c = 0.618. Hence, the geometrical mechanism responsible for the destruction is the existence of heteroclinic connections which obstruct the possible curves. Anyway, there are invariant objects with ρ in the above mentioned interval. It is proved that they should be at the outer part of the manifolds of the hyperbolic periodic orbit with ρ = 4/25 and at the inner part of the manifolds of the hyperbolic periodic orbit with ρ = 3/19. The heteroclinic intersections of these manifolds create gaps which forbid the existence of points of the invariant object in them. As a consequence, the invariant object which remains for some irrational ρ ∈ (3/19, 4/25) is a Cantor set [29, 30]. Points which were located inside an IC for c = 0.618 and, hence, without possible escape, can now, for c = 0.63, ﬁnd a gap of the Cantor set and escape under iteration. It looks like some random process and, certainly, the probabilities are related to the size of the gaps in the Cantor set.

3.2.2 The standard map Looking at the right-hand plot in Figure 3.2 we clearly see the period-5 islands around period-5 elliptic points and, as already said, we can guess the existence of period-5 hyperbolic points. We also see ICs close to the island, some of them inside, which have ρ > 1/5, and others outside, which have ρ < 1/5. If instead of iterations under Fc we iterate using Fc5 we will check that the inside curves turn

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a little clockwise and the outside ones turn a little counterclockwise. We can ask: what happens for an APM if we have two ICs turning by iteration a small amount in opposite directions? This is the contents of the so-called last geometric theorem by Poincar´e. Between the two curves, invariant under a map M , there should appear ﬁxed points, generically isolated and alternatively elliptic and hyperbolic. Typically, one point of each type appears. But if the map is the q-th power of some other ˜ , with rotation number p/q, (p, q) = 1, then there are q ﬁxed points of each map M ˜. type under M , which are q-periodic under M The structure of the islands is reminiscent of the phase portrait of a pendulum, whose Hamiltonian is H(x, y) = y 2 /2 + cos(x) using suitable coordinates. From a quantitative point of view (the width of the islands) we recall that the maximal distance between upper and lower √ separatrices in a pendulum with Hamiltonian H(x, y) = y 2 /2 + δ cos(x) is 4 δ. But we keep the presentation in the scaled version, i.e., with the coeﬃcient of the cosine equal to 1. The equations are x˙ = y, y˙ = sin(x). One can think of a discrete model which, in the limit, behaves as the pendulum. The simplest approach would be to use an explicit Euler method with step h, which gives the map (x, y) → (x + hy, y + h sin(x)). Unfortunately, that map is not an APM, but can be made symplectic using a symplectic Euler method: (x, y) → (¯ x, y¯), y¯ = y + h sin(x), x¯ = x + h¯ y. If we do not like to have the parameter h in both variables, we simply replace hy by a new variable z, rename z again as y, introduce k = h2 , and we obtain

x¯ = x + y¯ x , (3.10) → SMk : y¯ = y + k sin(x) y a popular map known as a standard map [7]. It is clear that we can look at the variables (x, y) in S × R or in T2 . It has ﬁxed points located at (0, 0), hyperbolic, and at (π, 0), elliptic, that we denote again as H and E. Figure 3.8 displays the phase portrait (in T2 ) for k = 0.5 and k = 1. On the left-hand plot it is hard to see that the stable and unstable manifolds of H do not coincide. A study like the one in the invariant manifolds part of Section 3.2.1 reveals similar properties. But the strongest diﬀerence between these plots is that in the left one there exist rotational ICs, that is, ICs going from the left vertical boundary to the right one (in this representation; in fact these boundaries are identiﬁed). These curves are absent in the right plot. Hence, if we consider the map in S × R, there is no obstruction to the dynamics in the y direction for k = 1. There are points with an initial value y ∈ [0, 2π) whose iterates can go arbitrarily far away in the y direction (despite the fact that for that value k = 1 will require many iterates). The critical value up to which one has rotational ICs is the so-called Greene’s critical value √ kG ≈ 0.971635; see [19]. The “last” rotational IC which is destroyed has ρ = ( 5 − 1)/2, the golden mean. This is not a surprise; it is the number in (0, 1) with best Diophantine properties. The obstruction method using hyperbolic

184

Chapter 3. Dynamical Properties of Hamiltonian Systems 6.28319

6.28319

4.71239

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Figure 3.8: Phase portrait of (3.10). Left: for k = 0.5, still quite well ordered. Right: for

k = 1, already with a big amount of chaos. Beyond the main elliptic island around E one can see several islands in both cases. The largest chaotic zone appears around the invariant manifolds of H.

periodic orbits with rotation numbers of the form Fn−1 /Fn and Fn /Fn+1 , Fn being the n-th Fibonacci number, plus a suitable extrapolation, allows us to determine kG accurately. Note also that for k > kG but close to kG , the rotational IC with √ ρ = ( 5 − 1)/2 is replaced by a Cantor set with “small holes”. This supports the claim about the large number of iterates needed to have y far away from the initial location. Renormalization theory [26, 27] provides the framework to understand those things in detail. For large values of k the standard map has interesting statistical properties. But they can be strongly aﬀected by the role of the islands, see [32]. On the other hand, the Hamiltonian H(x, y) = y 2 /2 + cos(x) can be replaced by more complex ones to obtain generalized standard maps. Adding terms in y 3 and y cos(x) allows us to explain the asymmetry which can be seen in Figure 3.2, right between the inner part and the outer part of the islands and the related inner, and outer splittings of the manifolds of the associated periodic hyperbolic points [52], in contrast with the symmetries of a pendulum. Replacing y 2 /2 by −by + y 3 /3 allows us to reproduce a limit ﬂow of the meandering curves, as shown in Figure 3.4, and other more complicated changes give rise to labyrinthine ICs with funny shapes [43].

3.2.3 Return maps: the separatrix map A useful device to understand the dynamics when some hyperbolic invariant object A has orbits homoclinic to it are the return maps. Assume that we have an initial point in a given domain D close to a point homoclinic to A. Then it approaches A under iteration, close to WAs , and after the passage near A moves away, close to WAu , and returns to D. Can we describe how the return is produced?

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To illustrate with an example we have used a modiﬁed H´enon–Heiles potential. In a pioneer example H´enon and Heiles in 1964 used a Hamiltonian with two d.o.f. (a model of the motion of a star in a galaxy with cylindrical symmetry) [21]. The Hamiltonian they derived is HH(x, y, px , py ) = (x2 + y 2 + p2x + p2y )/2 + x3 /3 − xy 2 ,

(3.11)

and a careful study of the behaviour of nearby orbits of the system (3.11) lead to the detection of chaotic motion, giving evidence of the lack of integrability, a fact that was proved theoretically later and that was relevant to face integrability problems from an algebraic point of view; see [34] and references therein. Later on the family with Hamiltonian HHc (x, y, px , py ) = (x2 + y 2 + p2x + p2y )/2 + cx3 − xy 2 was introduced, and the case c = 0, HHc=0 (x, y, px , py ) = (x2 + y 2 + p2x + p2y )/2 − xy 2 ,

(3.12)

that we shall use as illustration, presents some interesting particularities. Like many other simple models it has a symmetry with respect to simultaneous change of sign of y and t. One can ﬁx the value of the energy and use y = 0 as a Poincar´e section. The Poincar´e map P has a ﬁxed point H which corresponds to a hyperbolic periodic −1 orbit of (3.12). The invariant manifolds on HHc=0 (0.115) are shown in Figure 3.9. There exist homoclinic points and the symmetry implies that the upper branch of s u WH can be obtained from the lower branch of WH . 0.4

0.4 J

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Figure 3.9: Left: the invariant manifolds (W u in red, and W s in blue) of the hyperbolic simple periodic orbit of the modiﬁed H´enon–Heiles Hamiltonian located inside the domain of admissible conditions on the Poincar´e section y = 0, for the level of energy h = 0.115. The variables displayed are (x, px ). Right: a magniﬁcation of the upper part showing the location of sections I, J, K, and L mentioned in the text. The periodic orbit appears marked as H on the section.

Our goal is to describe the return to a suitable domain D as a model for a general setting. In Figure 3.9 on the right there is a homoclinic point in the segment

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market as I, whose image under P is the segment marked as J. A suitable domain can be a strip around the parts of the manifolds between I and J. Note that in the present case, due to the symmetry, we consider in Figure 3.9 (right) only the upper part in the (x, px ) variables. It can happen that, after passage near H, a point moves to the lower part. Hence, it is convenient to consider D as the union of two strips, symmetric the one with the other, and that we can denote as D+ and D− , according to the sign of px , and deﬁne D = D+ ∪ D− . In general, there is no symmetry and then D− is not obtained from D+ by symmetry, and even some of the branches of the manifolds can escape, as it happens for (3.5). u If we take the part of WH from H to the homoclinic point which appears s in J, followed by the part of WH between the homoclinic and H, and add the symmetric part (on px < 0) we have a ﬁgure-eight pattern which appears in many problems. This occurs, e.g., in the manifolds of the hyperbolic ﬁxed point of (3.10). Looking at the map in S × R in suitable coordinates, one has also a ﬁgure-eight pattern [53]. s First we assume that a point is located in D+ below WH (the line in blue). After passing close to H it will return to D+ . We follow an elementary method to ﬁnd the return map. If the splitting is small enough we can assume that the upper branches of the manifolds are coincident and consider a nearby integrable map in the domain bounded by the branches of the manifolds. Let ϕt be the ﬂow of a Hamiltonian v.f. with one d.o.f. in (x, px ) with Hamiltonian H such that ϕ2π coincides with this integrable map. In particular, points in I move to points in J under ϕ2π , and we can redeﬁne the strip D+ as the set ∪t∈[0,2π] ϕt (I). The s,u manifolds WH under that Hamiltonian v.f. coincide and form the separatrix of H. As additional variable in D+ , transversal to that separatrix, we take the level h of H, assuming H is positive inside the separatrix and equal to zero on it. For concreteness, let us denote as λ the dominant eigenvalue of the diﬀerential of the map ϕt=1 at H. It is clear that the dominant eigenvalue for ϕt=2π , close to the one of the initial map, say μ, is λ2π and that for the Hamiltonian v.f. is log(λ). If λ is close to 1 then log(λ) will be close to zero. For simplicity, we denote log(λ) as λ∗ . In terms of the dominant eigenvalue of the initial map one has λ∗ ≈ log(μ)/(2π). As all the orbits in the domain bounded by the separatrix are periodic under the ﬂow, when a point in D+ returns to it, it has the same value of h. But t has changed by the period, which behaves like c − log(h)/λ∗ , where c is a constant (essentially equal to the time to go from section L to section K). The map would be (t, h) → (t + c − log(h)/λ∗ (mod 2π), h). Now we return to our original map. The only change is due to the lack of s u coincidence of WH and WH . If we consider the variable h deﬁned as an energy u s with respect to WH , when continuing the motion close to WH the energy should be considered with respect to that manifold. There is a jump in energy due to the splitting. Note that, using a normal form around H, it is possible to deﬁne in a natural way an energy in a neighbourhood of this point, and to transport that

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187

function along both manifolds, by backward or forward iteration, see [11], and also [50] in a general context. The two energies do not coincide in D+ . The diﬀerence is the jump just mentioned. Let us denote it as s(t) (it has a weak dependence on h that we neglect). The simplest expression for s(t) is a sinusoidal oscillation u s ε sin(t) which measures the location of WH with respect to WH . Then the return map becomes

∗ ¯ t t¯ = t + c − log(h)/λ (mod 2π) . (3.13) → ¯ = h + ε sin(t) h h ¯ = 0 because then the point is in W s . It is clear that the map is not deﬁned if h H On the other, hand we have not considered the case h < 0. Then the process is similar, but we land on the lower domain D− . Beyond the variables (t, h) one has to consider a sign σ equal to ±1 in D± . Using also the sign and renaming the variables as ξ, η, with ξ ∈ [0, 2π) and η small, the map (3.13) becomes ⎛ ⎞ ⎞ ⎛ ξ ξ¯ = ξ + c − log(|¯ η |)/λ∗ (mod 2π) ⎠, η¯ = η + ε sin(ξ) SepM : ⎝ η ⎠ → ⎝ (3.14) σ ¯ = σ × sign(¯ η) σ a map known as a separatrix map. In a general case the jump ε sin(ξ) is replaced by a function s(ξ). In the asymmetric cases one uses diﬀerent jump functions s± (ξ) according to σ. The parameter ε, related to the size of the jump or splitting has, typically, exponentially small upper bounds as a function of some physical parameter, like the energy in the case of system (3.12). But in other cases, if the dominant eigenvalue at H tends to a constant λ0 > 1 when some parameter γ tends to zero, it can be, simply, a power of γ. For simplicity, we concentrate on the symmetric case and to points passing only through D+ . Then σ = 1 and we discard it in (3.14). Now we assume that η is ¯ 0) ≈ η ) = log(η0 )+log(1+ ζ/η close to some ﬁxed value, η0 , write η = η0 +ζ and log(¯ ¯ ¯ log(η0 ) + ζ/η0 , keeping only linear terms in ζ. This is a good approximation if ¯ 0 is small. If we assume, also, that λ is close to 1, then λ∗ is small. In the ζ/η ¯ ζ + ε sin(ξ)), where (ξ, ζ) variables the map becomes (ξ, ζ) → (ξ¯ = ξ + c1 + b1 ζ, ∗ ∗ c1 = c − log(η0 )/λ , b1 = −1/(η0 × λ ), and we do not write explicitly that ξ is taken mod 2π. Finally, deﬁne new variables u = ξ, v = c1 + b1 ζ and the map becomes

u¯ = u + v¯ u . (3.15) → v v¯ = v + b1 ε sin(u) Comparing (3.15) with (3.10) we see that they are identical if we set k = b1 ε = −ε/(η0 × λ∗ ). Therefore, we can expect to ﬁnd invariant curves in the separatrix map at a distance η0 > ε/(kG × λ∗ ) from the location of the invariant manifolds in D+ . A similar reasoning applies in the outer part, when the invariant curves make the full turn around the ﬁgure eight. This gives also an estimate of the width of

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Chapter 3. Dynamical Properties of Hamiltonian Systems

the zone with chaotic dynamics around the split manifolds. The estimate is quite realistic if ε is very small and λ is close to 1. This occurs, for instance, in a case like the Hamiltonian (3.12) because then, on a level of energy h, λ∗ and ε are, respectively, of the order of h and exponentially small in h. See [47] for a study of several cases, with diﬀerent number of d.o.f., either resonant or not. As ﬁnal comments in this subsection one has to add that it is very important to derive return maps in higher dimensions, like Hamiltonian systems with ≥ 3 d.o.f. or symplectic maps in dimension ≥ 4. But the formulas that one obtains can be far from simple, due to quasiperiodicity and resonances. To derive, from these return maps, bounds on the distance at which one can ﬁnd invariant tori, speed of diﬀusion, etc, is an open problem. See slides (J) for other open problems associated to some classes of global bifurcations.

3.3 Some theoretical results, their implementation and practical tools In this section we recall some general results and also provide tools to make them explicit.

3.3.1 A preliminary tool: the integration of the ODE, Taylor method and jet transport In the case of an analytic Hamiltonian (or general) v.f. like x˙ = f (t, x), x(t0 ) = x0 , (t0 , x0 ) ∈ Ω ⊂ R × Rn or Ω ⊂ C × Cn , one should use integration methods of the initial value problem for ODE. For instance, having in mind to compute Poincar´e iterates. A quite convenient method is the Taylor method. That is, to obtain the Taylor expansion x(t0 + h) for suitable values of h. If x(t0 + h) has components xi , i = 1, . . . , n, we look for a representation

xi =

N

(s)

ai hs ,

(3.16)

s=0

for suitable N, h, and use it as a one-step method. For further reference we denote (s) as a(s) the vector with components ai . The point is how to compute the coeﬃcients of the expansion in an easy way to high order. For a very large class of functions the evaluation of f can be split

3.3. Some theoretical results, their implementation and practical tools

189

into simple expressions e1 e2

= = .. .

g1 (t, x), g2 (t, x, e1 ),

ej

= .. .

gj (t, x, e1 , . . . , ej−1 ),

em f1 (t, x)

= = .. .

gm (t, x, e1 , . . . , em−1 ), ek1 ,

fn (t, x)

=

ekn .

Each one of the expressions ej contains a sum of arguments, a product or quotient of two arguments or an elementary function (like sin, cos, log, exp, √, . . .) of a single argument. The basic idea is to compute in a recurrent way the power series expansion (up to the required order) of all the ej . The gj have to be seen as operations with (truncated) power series. Hence, we can proceed as follows: (i) Input: t and the components of x0 , that is, the coeﬃcients of order zero in (3.16). (ii) Step s, s ≥ 0: from the arguments of gj up to order s we obtain the order s terms of ej ; in particular for fj (t, x), which gives the order s + 1 for xj (dividing by s + 1). This is repeated up to the required value of N . (s) (iii) The values of N, h can be selected so that the truncation error s>N ai hs is bounded, for every component, by some small ε negligible in front of the (unavoidable) round oﬀ error. Under reasonable assumptions, like c1 γ s ≤ ||a(s) || ≤ c2 γ s , 0 < c1 < c2 , (which implies radius of convergence ρ = 1/γ) in the limit when ε → 0, say ε = 10−d, with d large, one can take h such that the last term satisﬁes ||a(N ) ||hN = ε. It turns out, concerning eﬃciency, that the optimal value of h tends to ρ×exp(−2) (independently of the equation, and where ρ refers to the radius of convergence around the current point x0 ), and N ≈ d log(10)/2 when ε → 0. To carry out step (ii) above is immediate for arithmetic operations. As an example for elementary functions weconsider the case of powers, that we should s use to integrate (3.1). Let s≥0 us t , u0 = 0, α ∈ R, and we want to u(t) = α s compute v(t) = u(t) = s≥0 vs t . Then, v0 = uα 0,

vs = −

s−1 1 vk us−k [k − α(s − k)], su0 k=0

for s > 0, the determination being ﬁxed by the one used for v0 . This follows easily from v(t) = u(t)α by taking logarithms and diﬀerentiation with respect to

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t. Similar recurrences can be obtained for any elementary function. If f contains special functions (e.g., Bessel functions) it is enough to add the ODE satisﬁed by these functions to the system to be integrated. Computing to order N has a cost O(N 2 ). This is true for the most expensive elementary operations and functions, and it is the basis of the optimal estimates given above, see [22] and slides (A). In the autonomous case, to obtain the image of a point for a Poincar´e map P through a section Σ given by g(x) = 0 when g changes from < 0 to > 0, assume that we have a time t∗ such that g(ϕt∗ (x0 )) < 0 and g(ϕt∗ +h (x0 )) > 0, for the current value of h. Finding P(x0 ) reduces to solving a 1-dimensional equation, g(ϕt∗ +δ (x0 )) = 0, for the variable δ. This is easily done by using Newton’s method. Assume now that we look for a periodic solution. It can be written as a ﬁxed point of a Poincar´e map: G(x0 ) = P(x0 ) − x0 = 0, for x0 ∈ Σ. Again this can be solved by Newton’s method, but this requires that one knows the diﬀerential map DP(x0 ). To this end we integrate, together with the v.f. f , the ﬁrst order variational equations A˙ = Df (ϕt (x0 ))A, A(0) = Id. There are two points to take into account, see [40]: (i) The admissible variations of x0 should be conﬁned to the tangent space to Σ at that point. Furthermore, if the system has ﬁrst integrals, like in the Hamiltonian case, this gives additional constraints for x0 and the admissible variations if we ﬁx the levels of these integrals. (ii) The return time to Σ depends on the initial point. If instead of leaving from x0 we leave from x0 + ξ, ξ being an arbitrarily small admissible variation, the landing time in Σ has to be corrected by terms O(||ξ||). This is relevant to computing DP(x0 ). In some cases (see Subsection 3.3.4) we can be interested in having an approximation of the Poincar´e map not restricted to ﬁrst order terms in the variations of x0 ∈ Σ, but to higher order: we would like to have the Taylor expansion of P(x0 + ξ) to some given order in ξ. To this end, one can integrate the higher order variational equations, restrict the domain of deﬁnition to Σ and to the levels of the current ﬁrst integrals, or proceed in a diﬀerent, easier, way, using jet transport, described along the following lines. This can be also applied to obtain the image of a neighbourhood of a point x0 under ϕt , to see how it depends on parameters (useful to analyze bifurcations), etc. Assume the initial conditions are x0 + ξ, where ξ are some variations and we want to obtain ϕt (x0 + ξ) at order m in ξ. It is enough to replace all the operations described above to compute ej , in order to obtain the coeﬃcients in (3.16), done with numbers, by operations with polynomials in ξ up to order m. This applies to arithmetic operations, elementary functions, special functions, etc. Hence, instead of the vectors as of numerical coeﬃcients in (3.16) we deal with tables containing the numerical coeﬃcients, up to order m, of n polynomials in the ξ variables.

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191

If we return to the case of the Poincar´e map, we had to solve g(ϕt∗ +δ (x0 )) = 0, for the variable δ. Now δ will depend on ξ, but this is not a problem for Newton’s method. We simply apply it by replacing numbers by polynomials in ξ. We remark that the jet transport can be implemented in an eﬃcient way. It is also possible to produce rigorous estimates of the tails at every step, and to obtain intervals which contain the correct values of all the coeﬃcients. This allows us to convert a purely numerical simulation into a Computer Assisted Proof (CAP). See, e.g., [24].

3.3.2 Normal forms To study many systems, a useful trick is to try to reduce them to an expression as simple as possible, according to the topics of interest. If we study a discrete map around a ﬁxed point, it would be nice to be able to reduce it to a linear map. In general, this is not possible. Furthermore, we can be interested also in the dependence with respect to parameters, to analyze possible bifurcations. For concreteness we face a Hamiltonian in n d.o.f., in Cartesian coordinates, around a ﬁxed point (located at the origin) that we assume totally elliptic: the . , n. In canonically conjugate variables (xi , yi ), eigenvalues are exp(±i ωj ), j = 1, . . i = 1, . . . , n, we write it as H = k≥2 Hk , where Hk denote the homogeneous n 2 2 terms of order k and H2 = i=1 ωi (xi + yi )/2. In principle, we try to make a change of variables to cancel the terms Hk , k > 2. To keep the Hamiltonian character of the v.f. we shall use canonical transformations. These can be easily obtained as the ﬂow of an auxiliary Hamiltonian, G, with respect to an auxiliary time s until, say, s = 1. If you do not want to use a “so big time s = 1” simply scale (x, y) → ε(u, v), divide the Hamiltonian by ε2 obtaining H2 (u, v) + εH3 (u, v) + ε2 H4 (u, v)+· · · , and then the ﬁnal value of s will be ε. But this is equivalent to the previous approach. What makes the change close to the identity is the smallness of (x, y), not the fact of using s = 1. As we want to cancel, ﬁrst, the terms in H3 , weshall represent G also as a sum of homogeneous parts, starting at order 3, G = k≥3 Gk . To transform the function H under the change we write dH/ds = {H, G}, where n ∂H ∂G ∂H ∂G − {H, G} = ∂xi ∂yi ∂yi ∂xi i=1 denotes the Poisson bracket. Note that the bracket of homogeneous polynomials of degrees d1 and d2 has degree d1 + d2 − 2. Higher order derivatives are obtained by doing, successively, the Poisson bracket with G once and again. Trying to cancel (if it is possible to cancel) the terms Hk , k ≥ 3, we determine the homogeneous parts Gk . But it turns out that to obtain these parts it is much simpler to use complex coordinates. We introduce

1 xi 1 i qi = √ , i = 1, . . . , n. yi pi i 1 2

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Chapter 3. Dynamical Properties of Hamiltonian Systems

Then H2 becomes

n j=1

i ωj qj pj . The transformed Hamiltonian is

ϕG s=1 (H) = H + {H, G} +

1 1 {{H, G}, G} + {{{H, G}, G}, G} + · · · 2! 3!

(3.17)

Assume we have determined Gj , j < m, and we want to cancel all the possible terms to order m in (3.17). There are terms or order m in (3.17) which come from known and that we denote, together, Hm or involving Gj , j < m, which are already as Km . For deﬁniteness, assume Km = a,b,|a|+|b|=m Ka,b q a pb , where a denotes a n multiindex with n non-negative components ai , |a| = i=1 ai , and q a = Πni=1 qiai , b as usual. Similarly for unknown part comes from Gm , that we b and p . The only also write as Gm = a,b,|a|+|b|=m Ga,b q a pb and we would like to have 0 = {H2 , Gm } + Km =

a,b,|a|+|b|=m

i (ω, b − a)Ga,b q a pb +

Ka,b q a pb ,

a,b,|a|+|b|=m

(3.18) n where (ω, b−a) denotes the scalar product j=1 ωj (bj −aj ). As Ka,b is known, one easily determines Ga,b , provided (ω, b − a) = 0. But it is clear that if bj = aj for all j, then the term Ka,a must be left on the transformed Hamiltonian, independently of ω. These are called the unavoidable resonances which appear at even orders. Furthermore, if ω is resonant, i.e., there are integers cj , j = 1, . . . , n, such that (ω, c) = 0, other terms should be kept in the transformed Hamiltonian when b − a = c. These are the additional resonant terms. The normalization process can be continued to any order. But, in general, unless the Hamiltonian is integrable, the formal normal form is not convergent. One can expect that it belongs to some Gevrey class (see the invariant manifolds part in Section 3.2.1), but I am not aware of concrete general results in that direction. After we have transformed the Hamiltonian up to order M , we can skip the terms of higher order and denote the contribution up to order M as HN FM , the normal form to order M . We recall that a Hamiltonian system with n d.o.f. is said to be integrable (in the Liouville–Arnold sense) if there exist n ﬁrst integrals, Fj , j = 1, . . . , n, an involution, {Fi , Fj } = 0, and functionally independent almost everywhere. If ω is non-resonant then the HN FM is integrable, because one can take Fj = qj pj , ∀j. Now consider the resonant case. By construction, {H2 , HN FM } = 0 and, therefore, except in the degenerate case in which they are not independent, if n = 2 one has HN FM integrable. In general this is not true if n > 2. The system can be far from integrable even in a small vicinity of a totally elliptic point. But it can take a long time to have numerical evidence of the existence of chaos, even if it occurs for most of the initial conditions. A celebrated theorem by Arnold says that, for an integrable system, if the set of points in the phase space corresponding to ﬁxed values of the ﬁrst integrals F1−1 (c1 ) ∪ F2−1 (c2 ) ∪ · · · ∪ Fn−1 (cn ) is compact, then it is an n-dimensional torus

3.3. Some theoretical results, their implementation and practical tools

193

Tn . Around a given torus one can introduce the so-called action-angle variables (I, ϕ), I ∈ Rn , ϕ ∈ Tn . The integrable system can be written, then, as depending only on I: H = H0 (I), the integration is elementary and the frequencies on the given torus have the expression ωj = ∂H0 /∂Ij |F =c , j = 1, . . . , n. If the system is perturbed to H = H0 (I) + εH1 (I, ϕ) we can study how the properties of H0 (I) change under the eﬀect of the perturbation. See Subsections 3.3.3 and 3.3.5 in this direction. But we want to point out that it is also possible to try to produce a normal form for the perturbed Hamiltonian around the given torus if the frequencies on it, ωj , satisfy a non-resonant condition. This can push the perturbation to higher order in ε, making easier the applicability of general results. Up to now we have considered, around a ﬁxed point, the totally elliptic case. ne 2 2 contains some hyperbolic part H = If the quadratic term H 2 2 i=1 ωi (xi +yi )/2+ n λ x y , one can use similar ideas to obtain approximations of the central j=ne +1 j j j manifold and of the Hamiltonian reduced to it. We return to this in Subsection 3.3.4.

3.3.3 Stability results: KAM theory and related topics There is a natural generalization of the idea of twist map to higher dimension. Consider a map T deﬁned, in suitable coordinates, in a product of n annuli, with radii ri ∈ (rd,i , ru,i ), 0 < rd,i < ru,i , i = 1, . . . , n, of the form T (r, α) = (r, α+a(r)), where r ∈ R = Πni=1 (rd,i , ru,i ) has components r1 , . . . , rn , α ∈ Tn and a is a map from R to Rn which can be denoted as translation. The map T is an integrable symplectic map, and R × Tn is foliated by tori invariant under T . Nothing else but what we saw for (3.7) in the part about ICs in Section 3.2.1. The diﬀerential of the translation with respect to the radii, Dr a(r) is known as torsion. Then the KAM theorem for symplectic maps has the following statement, completely analogous to Theorem 3.2.1. Theorem 3.3.1. Consider a perturbation Fε = T + εP of the integrable symplectic map T in R × Tn , and assume that for r = r∗ the vector a(r∗ ) satisﬁes a DC, that the torsion is non-degenerate and ε is small enough. Then the map Fε has also invariant tori in Tn , close to r = r∗ , and on them the action of Fε is conjugated to the one of T on r = r∗ , that is, a translation by a(r∗ ). In the present case, the DC is slightly diﬀerent from the one in (3.3). Beyond the translations ai (r), i = 1, . . . , n, one has to add the value 1, as it is obvious thinking on the suspension. So, it reads as n ki , ai ) + k0 ≥ b|k|−τ , ∀k ∈ Zn+1 \ {0}, ( i=1

where k denotes now (k1 , . . . , kn , k0 ). The role of the DC, the non-degeneracy of

194

Chapter 3. Dynamical Properties of Hamiltonian Systems

the torsion, is analogous to the twist condition, and the smallness of ε plays the same role. A result similar to Theorem 3.3.1 holds in the case of Hamiltonian systems. Theorem 3.3.2. Let H0 (I) be an integrable Hamiltonian, for which there exist invariant tori, and assume that for some given torus, labelled by I ∗ , the frequencies ω(I ∗ ) = ∂H0 (I)/∂I|I=I ∗ satisfy a DC (in the sense of (3.3)) and are nondegenerate, so that the diﬀerential ∂ω/∂I|I=I ∗ is regular. Then if ε is small enough, a perturbed Hamiltonian H(I, ϕ) = H0 (I) + εH1 (I, ϕ, ε) has a nearby invariant torus with the same frequencies. These results usually do not give estimates on how small ε should be or, if any, they are very pessimistic. However, normal form techniques (see Subsection 3.3.2) can help to start the iterative process in a very good approximation, so that the diﬀerence with the initial guess and the true torus, if it exists, is suﬃciently small. For the eﬀective computation of invariant tori there exist diﬀerent methods. A quite classical method is the Lindstedt–Poincar´e (LP) method. In principle, it is formal because one looks for the invariant tori without paying too much attention to the DC (despite the fact that this can also be implemented). Assume that we look for 2D tori around a totally elliptic point (assumed to be located at the origin) in a Hamiltonian system with n = 2 d.o.f. Let ω1 (0), ω2 (0), be the frequencies at the ﬁxed point. The linear system will have, for the q, p variables, a representation as linear combinations of cos(ω1 (0)t + ψ1 ) and cos(ω2 (0)t + ψ2 ), where ψ1 , ψ2 represent some phases, and these terms have amplitudes α1 , α2 . Due to symmetries and the freedom to select the origin of time, the phases for the diﬀerent variables can be put in simple form. We wish to satisfy the equations q˙ = ∂H/∂p, p˙ = −∂H/∂q by expanding in powers of the amplitudes α1 , α2 and integration of the coeﬃcients of these powers with respect to time. However, it turns out that at some order we can ﬁnd on the right-hand side of the equations terms which are not purely quasiperiodic, i.e., they are constant. The solution consists in allowing the frequencies to depend also on the amplitudes. So ωi = ωi (0) + j1 ,j2 ci,j1 ,j2 αj11 αj22 , i = 1, 2, and a suitable choice of these ci,j1 ,j2 coeﬃcients cancels the constant terms. An often used method is based in writing the coordinates of the points of the unknown torus as Fourier series in some angles, and then imposing the invariance conditions. For concreteness we consider the case of symplectic maps. The ﬂow case can be reduced to this one via a Poincar´e map. Assume that we look for a d-dimensional torus in which the dynamics is conjugated to θ → θ + α, for θ ∈ Td and a translation vector α ∈ Rd satisfying the DC. Let x be the coordinates in the phase space and F the discrete map. The invariance condition is F (x(θ)) = x(θ + α).

(3.19)

It is clear that one has freedom to select the origin of the angles θi and that eventual symmetries can reduce the number of coeﬃcients to be determined.

3.3. Some theoretical results, their implementation and practical tools

195

To begin with the process, we can assume that we have obtained some approximation by direct numerical simulation, or that we start near a ﬁxed (or periodic) point and use the linear approximation or an approximation obtained by an LP method. If we are interested in a family of invariant tori, one can use continuation methods, but taking into account that the values of α should satisfy the DC. Hence, there will be gaps in the family, despite the fact that they can be very small in some cases. Let c denote, generically, the coeﬃcients of the Fourier expansion, truncated at a suitable order. From a grid of values of θ one can obtain initial values of x. They are mapped to F (x) and the images can be Fourier analyzed to obtain the new Fourier coeﬃcients cˆ. Let L be the action of the translation by α on the initial Fourier coeﬃcients. According to (3.19), we should require L(c) − cˆ = 0. This is the equation that follows from (3.19) and has to be solved, usually by Newton’s method. The diﬀerentials of the Fourier synthesis and analysis are elementary and the one of F can be obtained by computing DF (this can be, typically, the diﬀerential of a Poincar´e map). See [23] for an eﬃcient implementation with similar and extended ideas, which works even with a very large number of harmonics. The number of harmonics to be used depends on the shape of the torus. One can use in the grid in θ (and, therefore, in x) a number of points larger than the number of components of c. In that way one can check the behaviour of coeﬃcients in cˆ which have not been used as c coeﬃcients in the representation of the solution we search, and see if they can be neglected. Otherwise, one increases the number of harmonics. This can be done at successive iterations of Newton’s method in a dynamic way. It is also possible not to ﬁx α a priori and determine it together with the coeﬃcients c. Note that, in case α is close to resonant, one can have convergence problems. For other quite diﬀerent problems, like looking for invariant tori in PDE, this method requires a huge number of Fourier coeﬃcients if the discretisation dimension is large. Other methods, working directly in the phase space like the synthesis of a return map, see [39, 42], can give the desired results. There is a fact, concerning invariant tori and which applies also to the computation of some periodic orbits, which can produce diﬃculties. This is the instability present in partially normally hyperbolic tori or, in a simpler case, in linearly unstable periodic orbits. Given a point x, and assuming it approximately located in an invariant torus, the instability can produce that F (x) is far away from the torus. This produces convergence problems. The solution consists in using parallel shooting. Instead of taking a single Poincar´e section, say Σ, one can use several of them, say Σ0 = Σ, Σ1 , Σ2 , . . . , Σm−1 , and the corresponding partial Poincar´e maps: P1 : Σ0 → Σ1 ,

P2 : Σ1 → Σ2 ,

...

Pm : Σm−1 → Σ0 .

Hence, the full Poincar´e map can be written as P = Pm ◦ · · · ◦ P2 ◦ P1 . Then we look for Fourier representations in each one of the intermediate sections. This

196

Chapter 3. Dynamical Properties of Hamiltonian Systems

produces a much larger set of equations, but it has the advantage that each one of the partial maps Pj is much less unstable. In the case of highly unstable periodic orbits things are simpler. We only need one point in each intermediate section, say x0 ∈ Σ0 , x1 ∈ Σ1 , . . . , xm−1 ∈ Σm−1 . The conditions are simply P1 (x0 )−x1 = 0, P2 (x1 )−x2 = 0, . . . , Pm (xm−1 )−x0 = 0. The system to be solved is large but the diﬀerential has a simple block structure and the condition number is much better.

3.3.4 Invariant manifolds Another basic ingredient of the dynamics are the invariant manifolds. In contrast with the tori of maximal dimension, responsible for the regular behaviour, the invariant manifolds are, typically, responsible for the chaotic part of the dynamics. We comment ﬁrst on invariant stable and unstable manifolds of ﬁxed points of APM F . The components will be denoted as F1 , F2 . Assume a ﬁxed point is located at the origin with dominant eigenvalue λ > 1, and having an unstable linear subspace E u and a stable one E s . Then the u in unstable manifold Theorem ensures the existence of an unstable manifold Wloc u a neighborhood of the origin, invariant under F , tangent to E at the origin and such that for a point p on it, the iterates under F −1 tend to the origin. In fact, u remain on the neighborhood for all iterations. This is a only the points in Wloc u local result. Then the global unstable manifold W u is obtained by iteration of Wloc under F . A similar result gives the stable manifold, obtained by exchanging F and F −1 . In the analytic case, as we assume, the manifolds are analytic. Let u and s be local coordinates along the unstable and stable eigenvectors. For the linear map DF , the manifold W u is just s = 0. We can try to ﬁnd a representation of W u for F as the graph of a function: s = g(u) = j≥2 aj uj . The invariance condition reads F2 (u, g(u)) = g(F1 (u, g(u))). The coeﬃcients aj are determined in a recurrent way by identifying the left-hand and right-hand coeﬃcients of uj . An alternative representation of W u is the parametrization method. Let us use z as a parameter. In the linear case, a point with u = z is mapped to u = λz. Now it is not necessary to use coordinates adapted to the eigenspaces. If we use (x, y) as coordinates around the ﬁxed point and represent the parametrization as (p1 (z), p2 (z)), the invariance condition is simply F (p1 (z), p2 (z)) = (p1 (λz), p2 (λz)).

(3.20)

That is, we look for a conjugacy on the manifold between F and its linear part. We search now for the parametrization as p1 (z) = j≥2 aj z j , p2 (z) = j≥2 bj z j in (3.20). Note that the parametrization can be normalized so that the vector of coeﬃcients of order 1 has Euclidean norm equal to 1. As before, the coeﬃcients of order j > 1 are obtained in a recurrent way. This is the method used for many of the examples displayed before.

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197

A ﬁrst practical question, given a parametrization to order N (a similar question can be posed for the graph method), is up to which value of z, say zmax , one can use the representation. The idea is quite simple: given a tolerance ε we can compute the point B of parameter z and also the point A of parameter z/λ. One should have F (A) = B, according to (3.20). Hence, we can check up to which value of z one has ||F (A) − B|| < ε. This gives the admissible domain for z. Then, a fundamental domain F D is parametrized by z ∈ (zmax /λ, zmax ]. Any point on the manifold can be found as an iterate of a point in F D. A similar domain, with z < 0, has to be found for the other branch of the manifold. To obtain points in the manifold for z > zmax we simply divide the current parameter by λ as many times as required until a value less than zmax is obtained. Assume one has to divide k times. Then we compute the point of parameter z/λk and iterate it k times under F . In this way it is possible to reach points away from the ﬁxed one, to detect foldings of the manifold, to reach the vicinity of a homoclinic or heteroclinic point, etc. The selected values of z at which the computation is done can be chosen to satisfy conditions such as having the distance between two consecutive points in W u or the angle between three consecutive points below some preﬁxed values. Why do we need approximations beyond the linear one? The answer depends on the purpose. If we want to produce a long part of the manifold and, especially, if λ is close to 1, we can require many iterates. On the other hand, if F is not given explicitly but follows from a Poincar´e map, we need jet transport to have a local Taylor expansion. In any case, there is an optimal choice to obtain the “cheaper order” (cheaper can mean in terms of CPU time, of personal time, or a combination of both). If we are interested in locating a homoclinic point, and no symmetry is available for this, the problem reduces to ﬁnding two parameters, zu and zs , and well as two integers, ku and ks , to be used for the unstable and stable manifolds, respectively, such that F ku (pu (zu )) = F −ks (ps (zs )), where pu , ps denote the respective parametrizations. It is possible to ﬁnd suitable values of ku , ks and then to solve for zu , zs using Newton’s method. A similar method can be used for heteroclinic points, for tangencies, etc. The ideas are similar in higher dimension. One can look for d-dimensional invariant manifolds, d > 1, using either graph or parametric methods. This is specially necessary, for instance, if we look for an unstable manifold with quite diﬀerent eigenvalues. A low order representation will take the initial points along the direction of the maximal eigenvalue. Beyond using high order local expansions, to decrease the problem, one can use diﬀerent devices depending on the problem. To look for the invariant unstable manifold of an invariant curve in a symplectic 4D map, a parametrization using a parameter z, which measures the distance to the curve, and an angle θ along the curve are useful. The fundamental domain, in that case, is diﬀeomorphic to an annulus. See an example in Subsection 3.4.2 in a diﬀerent context, and another one in Subsection 3.4.5 concerning a family of invariant curves.

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The idea extends to any dimension with increasing complexity. See [2, 3, 4] for a nice global approach. A diﬀerent problem appears when we consider symplectic maps in dimension 4 (or higher) or problems reducible to them. Consider again the case of a ﬁxed point but assume that, together with an eigenvalue λ > 1 and its inverse, there is a couple of eigenvalues of modulus 1. They give rise to the centre manifold of the point. In general, when we consider a given neighborhood of the point, the manifold has some degree of diﬀerentiability which depends on the neighborhood. Furthermore, there is no uniqueness in general. The diﬃculty comes from the fact that the dynamics on that manifold is not known. It can contain, simultaneously, invariant curves, periodic points and chaotic zones. It is said to be a normally hyperbolic invariant manifold (NHIM) if the hyperbolicity normal to the manifold is stronger than the hyperbolicity that can be found inside the manifold. One can recur to normal forms to obtain an approximation of all the dynamics around the point and, in particular, the centre manifold. A similar idea is to use a partial normal form, see [42]. Assume we have a Hamiltonian 1 1 Hk (q1 , q2 , q3 , p1 , p2 , p3 ), (3.21) H = λq1 p1 + ω1 (q22 + p22 ) + ω2 (q32 + p23 ) + 2 2 k≥3

where, as usual, Hk denotes a homogeneous polynomial of degree k. We proceed as in the case of normal form above, but trying only to cancel all the terms such that the total degree in (q1 , p1 ) is equal to 1. Using again complexiﬁcation, as in the case of the normal form, for the couples (q2 , p2 ) and (q3 , p3 ), the current denominators to obtain the successive terms in the Hamiltonian G used to transform H are of the form (k1 − l1 )λ + i (k2 − l2 )ω1 + i (k3 − l3 )ω2 , with modulus bounded from below by |λ|, even if ω1 and ω2 are resonant. It is clear that, denoting the new variables as Q1 , Q2 , Q3 , P1 , P2 , P3 , if we set Q1 = P1 = 0 this is the desired centre manifold. Hence, setting these variables to zero we have a Hamiltonian with two d.o.f., which gives the reduction to the centre manifold of the initial Hamiltonian. The process is formal, there is no convergence in general, but one can obtain a good approximation in a suitable domain. One can check up to which distance of the ﬁxed point the approximation satisﬁes some tolerance condition. See [42] for an example around the collinear point L2 in the spatial circular restricted three-body problem.

3.3.5 Instability, bounds and detection In the case of a Hamiltonian with n ≥ 3 d.o.f., in principle, there is no way to avoid diﬀusion. The maximal dimensional tori have dimension n, that is, codimension n− 1 in a ﬁxed level of energy, and they do not separate the phase space. For instance,

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initial conditions as close as we like to L4,5 in the spatial circular restricted threebody problem (see the beginning of Section 3.4), which are totally elliptic ﬁxed points, can go far away from these points. But normal forms, or averaging, lead to the so-called Nekhorosev estimates [37], showing that one needs an extremely large time if one starts close enough to the libration point. See also [14] for a rather detailed approach. Similar things happen for (2n−2)-dimensional symplectic maps. Consider a perturbation H(I, ϕ) = H0 (I) + εH1 (I, ϕ, ε) of an integrable Hamiltonian H0 . The basic idea of the bounds is similar to the averaging Theorem 3.1.1, trying to cancel, around an arbitrary torus labelled by the action I ∗ , the dependence with respect to ϕ. But now the frequencies of the unperturbed Hamiltonian ω(I) = DH0 may not satisfy the DC and, on the other hand, the perturbation will produce that the frequencies change. Hence, the passage through resonances or through other frequencies not satisfying the DC is unavoidable. First one should examine what is the eﬀect of a resonance. We refer to Subsection 3.2.2 where we commented on the width associated√to a pendulum like structure. A perturbation O(ε) can give rise to variations O( ε) due to the presence of a simple resonance. This happens if the frequencies change, reach a resonance and then go away from it. But then one can put the following question. Assume that in the variation of some action there is a term, due to the perturbation, like I˙j = ε cos((k, ϕ)), where (k, ϕ) is a linear combination of the angles, and the related combination of the frequencies satisﬁes (k, ω) = 0. One expects that the frequencies will change with time and one will escape from resonance, but it can happen that the frequencies are locked at resonance up to order m for some dm+1 dk m > 0. That is, dt k (k, ω) = 0 for k = 0, 1, . . . , m, and dtm+1 (k, ω) = 0. Then, during a long time, the term cos((k, ϕ)) will be close to constant and the action can change by a large amount. If the locking occurs at all orders, the change in Ij will be O(εt). To prevent this locking is why Nekhorosev introduced the so-called steepness condition, which prevents the order of the locking exceeding a maximal value. Then one has the Nekhorosev result: under steepness of some order, the variation of the actions ||I(t) − I(0)|| does not exceed a bound O(εb ) during a time interval |t| < O(exp(cε−a )), where the positive constants a, b, c depend on the order of steepness and properties of H0 , assuming that the norm of H1 is bounded. Around a given point, or a given torus (in particular, a periodic orbit) it can happen that there are many KAM tori. The above description of the Nekhorosev estimates puts a bound on how fast escap from the vicinity of these tori can be. Typically, one refers to this fact as stickiness of the invariant tori. Perhaps the escape is so slow that it has no relevance during the time interval in which we are interested, or even during the period of validity of the model. This suggests that we introduce the concept of practical stability. Assume that the studied object has I = I ∗ . Then, for ﬁxed values of (ε, T ), where ε is moderately small and T is large, we say that there is (ε, T )-practical stability if there exists ρ = ρ(ε, T ), such that points with initial conditions at t = 0 satisfying ||I(0) − I ∗ || < ρ evolve with time

200

Chapter 3. Dynamical Properties of Hamiltonian Systems

satisfying ||I(t) − I ∗ || < ε for all t ∈ [0, T ]; that is, we require stability only for ﬁnite time. See [14]. Clearly, for any v.f. with Lipschitz constant L, (ε, T )-practical stability is found if ρ ≤ ε exp(−LT ), as follows from Gronwall’s Lemma. But this gives extremely small values of ρ, completely useless for any practical application. More realistic values would be ρ = 0.01 for ε = 0.02 and T = 109 , depending on the practical example in mind. See, e.g., [9] for a nice approach to KAM and practical stability simultaneously. Another relevant point is how to detect the existence of chaos and quantify it in a concrete example. There are many diﬀerent approaches. We comment on the Lyapunov exponents. To measure the instability properties of a ﬁxed point (of a continuous or discrete system) it is enough to look at the diﬀerential of the v.f., or of the map at that point. How to proceed for a general orbit? The idea is to look for the rate of increase (if any) of the distance between the orbits of nearby points. In the limit, this becomes the rate of increase of an initial displacement, ξ, under the diﬀerential of the iterates of the map or under the action of the ﬁrst order variational ﬂow. For concreteness we consider the case of discrete maps. Let x0 be an initial point on a manifold M on which it acts a map F , and let x1 = F (x0 ), . . . , xk = F (xk−1 ), . . . be the orbit of x0 . We can deﬁne, if it exists, log(||DF k (x0 )ξ||) , k→∞ k

Λ = sup lim ξ

(3.22)

where ξ is taken from the vectors with unit norm ||ξ|| = 1 in Tx0 M, the tangent space to M at the point x0 . One can prove that the limit in (3.22) exists for almost every x0 ∈ M and for almost every ξ ∈ Tx0 M, and it is known as maximal Lyapunov exponent. In the Hamiltonian case (or in the symplectic one) it is easy to prove that, for initial points in invariant tori of maximal dimension, the limit exists and is equal to zero. Typically, ||DF k (x0 )ξ|| behaves linearly in k in that case, which gives the desired limit. For generic unstable orbits one expects positive values of Λ. The geometrical reason is clear: every time that the iterates pass close to an hyperbolic object, the unstable component will increase at a geometric rate. For an integrable system, if, for instance, unstable and stable manifolds coincide, when returning near the hyperbolic object, this expansion is canceled due to the iterations which occur close to the stable manifold. But the existence of transversal homoclinic (or heteroclinic points) prevents this from occurring. One of the basic questions is how to have an estimate of the limit. In practice the number of iterations should be ﬁnite (and there is also the eﬀect of round oﬀ, which is another issue). A simple approach is to proceed to the computation in (3.22) using a diﬀerent presentation. Let us deﬁne the Lyapunov sums as follows. Let x0 , ξ0 be the initial point and vector, and set S0 = 0. Then, at the k-th iterate,

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201

we use the following algorithm: Sk = Sk−1 + log(||ηk ||). (3.23) Hence, we normalize the tangent vector after every step and add the log of the normalization to the current value of the sum S. It is clear that the limit slope of Sk , as a function of k, should coincide with Λ, as deﬁned in (3.22). Hence, we can proceed as in (3.23) and, from time to time (say, after mN iterates, m = 1, 2, . . .), ﬁt a line to three diﬀerent subsamples of the current sample (e.g., last 30%, last 50% and last 70%) and accept the average of the slopes as value of Λ if they diﬀer by less than a prescribed tolerance. Otherwise, keep iterating until the next multiple of N , provided this does not exceed a maximal value. A problem is that, in case Λ = 0, the convergence can be slow; for instance, log(k)/k is below 10−5 only for k ≥ 1, 416, 361. An alternative approach, which tends in a faster way to the limit and also smoothes out the oscillations due to the quasiperiodic eﬀects (in the case of orbits), can be found in [8]. One can look for the systematic use of that method in [25] for a family of 2D symplectic maps in S2 . Another idea, if one is interested only in deciding whether the orbit is regular or chaotic, is to stop computations and consider the orbit as chaotic if Sk exceeds some threshold. xk = F (xk−1 ),

ηk = DF (xk−1 )ξk−1 ,

ξk = ηk /||ηk ||,

3.4 Applications to Celestial Mechanics In this section we present several applications to illustrate theoretical and computational approaches to simple examples in Celestial Mechanics. One can have a look at slides (C), concerning the role of dynamical systems in Celestial Mechanics. Most of the applications deal with the restricted three-body problem (RTBP). We shortly recall it. The RTBP studies the motion of a particle P3 of negligible mass under the gravitational attraction of two massive bodies, P1 and P2 , of masses m1 and m2 , respectively. They are known as primaries or as primary and secondary. We assume that the primaries move in a plane along circular orbits around their centre of masses. We can normalize m1 +m2 = 1 and d(P1 , P2 ) = 1 and express the dynamics in a rotating frame (the so called synodical frame) with unit angular velocity. The problem depends on a unique parameter μ = m2 . In this frame P1 and P2 are kept ﬁxed at (μ, 0, 0) and (μ − 1, 0, 0). The equations of motion are x¨ − 2y˙ = Ωx , where Ω(x, y, z) =

y¨ + 2x˙ = Ωy ,

z¨ = Ωz ,

1 2 1−μ μ μ(1 − μ) (x + y 2 ) + , + + 2 r1 r2 2

(3.24)

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r12 = (x − μ)2 + y 2 + z 2 , and r22 = (x + 1 − μ)2 + y 2 + z 2 . The function J(x, y, z, x, ˙ y, ˙ z) ˙ = 2Ω(x, y, z) − (x˙ 2 + y˙ 2 + z˙ 2 ) is a ﬁrst integral, its value being known as Jacobi constant and it is usually represented as C. The related 5D energy manifolds are deﬁned as 2 1 ˙ y, ˙ z) ˙ =C (3.25) M(μ, C) = (x, y, z, x, ˙ y, ˙ z) ˙ ∈ R6 |J(x, y, z, x, and their projections on the conﬁguration space are known as Hill’s regions, bounded by the zero velocity surfaces (ZVS) (the zero velocity curves, ZVC, in the planar case). The problem has ﬁve equilibrium points (also known as libration points): (i) Three of them, say L1 , L2 and L3 , are collinear (or Eulerian) on the x-axis, of centre×centre×saddle type and, hence, they have a 4D centre manifold which contains the so-called horizontal and vertical periodic orbits of Lyapunov type (to be denoted as hpoL and vpoL ), invariant 2D tori and other periodic orbits (like the halo orbits, depending on the value of C), as well as chaotic regions. (ii) Two of√them, say L4 and L5 , are triangular (or Lagrangian) at x = μ − 1/2, y = ± 3/2, z = 0. The term μ(1 − μ)/2 in Ω is added to have C(L4,5 ) = 3. Let μj be the value of μ for which the ratio of frequencies in the plane, 1/2 (1 ± (1 − 27μ(1 − μ))1/2 )/2 , is j. The points are totally elliptic for 0 < √ μ < μ1 = (9 − 69)/18√and the 2:1, 3:1 resonances√(leading to instability) show up for μ2 = (45 − 1833)/90 and μ3 = (15 − 213)/30. Associated to the planar frequencies there are the so-called short and long period periodic orbits. The vertical frequency, giving rise also to a family of vpoL , is equal to 1.

3.4.1 An elementary mission around L1 First we consider the planar case. Assume that P1 and P2 are Sun and Earth, respectively. The distance between them, 1.5 × 108 km, and the period, 1 year, are scaled to 1 and 2π units, respectively, as said before and we take μ = 3.0404326 × 10−6 (it includes Moon’s mass). We want to carry out the following steps: (i) Compute a periodic orbit of the system, around the Earth, with a period of 1 day (a geostationary orbit) and check that it is close to circular. Call it PO1 . (ii) Compute some periodic orbits around L1 (of the hpoL family), which are symmetrical with respect to the x-axis. Check that they are unstable. We call them, in general, PO2 . (iii) Compute the left branches of the stable manifolds of the previous orbits until they reach some suitable value of x (e.g., x = −0.999).

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(iv) Now assume an spacecraft is moving in the “parking” orbit PO1 . At some point of the orbit we give an impulsion Δv, in the direction of the velocity at that point, with the goal of reaching a point of the stable manifold of one of the hpoL . Determine the hpoL which are reachable in that way from the parking orbit, at which place one should give the impulsion and which is the size Δv. This allows us to obtain an elementary approach to a space mission. Later, one can consider the eﬀect of perturbations of other bodies, the separate eﬀects of Earth and Moon, change to a non-planar target orbit, the fact that the target orbit is, approximately, quasiperiodic instead of periodic, to optimize with respect to fuel consumption and with respect to transfer time from departure to a vicinity of the target orbit, etc. For information about the methodology for the design and control of missions around libration points see [15, 16, 17, 18]. We detail the steps to ﬁnd the solution in the present example. Step 1: First we compute a periodic orbit around the Earth with period τ = 2π/366.25. We start with initial data (x0 , 0, 0, y˙ 0 ) and require ϕτ (x0 , 0, 0, y˙ 0 ) = (x0 , 0, 0, y˙ 0 ). In fact, it is much simpler to ask for the image at t = τ /2 to be of the form (x1 , 0, 0, y˙ 1 ), and then symmetry completes the task. We have two known data x0 , y˙ 0 and two conditions y1 = 0, x˙ 1 = 0. After a few attempts one can use Newton’s method to ﬁnd the solution x0 ≈ −0.999714471273, y˙ 0 ≈ 0.103463316596. One can check that the monodromy matrix has a double eigenvalue equal to 1 (as expected: energy preservation and time shift) and the other eigenvalues are exp(±αi ), α ≈ 0.034228998. The diﬀerence with respect to a circular orbit is less than 350 m. For further reference we denote this orbit as γ(t). Step 2: Now we face the hpoL around L1 . First we locate L1 by imposing Ωx = 0 as it follows from (3.24). Starting at x = μ − 1 + (μ/3)1/3 , Newton’s method converges quickly for μ small. Then we can compute the eigenvalues at that point, which turn out to be λ, λ−1 , exp(±ωi ), with λ ≈ 2.532659199, ω ≈ 2.086453579. Hence, the maximal eigenvalue of the nearby periodic orbits, when they tend to L1 , is exp(2πλ/ω) ≈ 2052.671203. This large instability suggests, again, that we look for the initial data for the hpoL on the Poincar´e section y = 0 for a ﬁxed x0 with x˙ 0 = 0, and leaving y˙ 0 as the only unknown variable. The condition to be satisﬁed is then that the next intersection with y = 0 (to the left of L1 ) should have x˙ = 0. This is easily solved by Newton’s method. From the half orbit we recover the full orbit by symmetry, the monodromy matrix and, hence, dominant eigenvalue and eigenvector. The instability becomes milder when the size increases. For instance, for the smallest orbit in Figure 3.10 on the left the dominant eigenvalue is 2050.987058, while the largest one is 923.004416. Standard continuation techniques are used to generate these orbits. Step 3: With the previously computed data it is simple to produce the left branches of the stable manifolds WPs,− O of the hpoL until they intersect the value x = −0.999.

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0.01

0.01

0.005

0.005

-0.005

-0.005

-0.01

-0.01 -0.994

-0.992

-0.99

-0.988

-1

-0.996

-0.992

-0.988

Figure 3.10: Left: some orbits in the family hpoL around L1 . The initial values of x,

on y = 0, are of the form xL1 + k × 10−4 for k = 1(1)22. Right: For some of the orbits, concretely for k = 6(4)22, we plot also the left branches of WPs Ok until they reach x = −0.999. In both plots the variables x, y are shown.

It is enough to use the linear approximation of the manifold in the Poincar´e section y = 0. An example is shown in Figure 3.10 to the right. To compute the manifolds 200 points have been taken in a fundamental domain, equally spaced in logarithmic scale. The intersections for the orbits with k = 8(2)22, i.e., for the indices ranging from 8 to 22 with step 2 (see Figure 3.10) are shown in red in Figure 3.11 on the right, using y, y˙ as variables. Step 4: The last step is how to reach WPs,− Ok for a given k leaving from the parking orbit. It is suggested to give an impulsion Δv from a given point γ(t∗ ) in the orbit, in the direction of the velocity γ(t ˙ ∗ ) at that point. The ﬁrst question is to compute what is the size of the new velocity. We simply require that the value of the Jacobi constant with this velocity equals the one of the target P Ok . Let |v| be the modulus obtained for this velocity. Then, Δv = |v| − |γ(t ˙ ∗ )|, and the ∗ components of the new velocity are proportional to the ones of γ(t ˙ ). This allows us to compute the trajectories ψ(t, t∗ ) leaving from the parking orbit until they reach x = −0.999. Depending on t∗ it can happen that ψ(t, t∗ ) reaches x = −0.999 or it goes ﬁrst far away to the left, spending too much time. These trajectories are skipped. A sample of the possible ψ(t, t∗ ) trajectories for several t∗ values is shown in magenta in Figure 3.11 on the left, where the parking and target orbit (with k = 14) are in red, and WPs,− O14 is shown in blue. Finally, on the right-hand part of Figure 3.11 we show, in the (y, y) ˙ variables, the information that has been obtained in x = −0.999: the intersections of WPs,− Ok for k = 8(2)22, in red, and the intersections of ψ(t, t∗ ) when one changes t∗ , for the Jacobi levels of P Ok , k = 10(4)22, in blue. The intersections of a given red curve with the corresponding blue one are the candidates for the transfer. The values of Δv are quite close. They range from 0.040286 for k = 10, to 0.041246 for k = 22 (i.e., impulsions ranging from 1.203 to 1.232 km/s).

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0.008 0.06

0.004 0.03

0 0

-0.004 -0.03

-0.008 -1.002

-0.998

-0.994

-0.99

-0.01

-0.006

-0.002

0.002

Figure 3.11: Left: The parking orbit and an example of a possible target hpoL (with k = 14), both in red, the branch WPs,− O14 in blue, and some of the possible trajectories ψ(t, t∗ ) departing from the parking orbit (see text) in magenta. Plot done using x, y variables. Right: The intersections of WPs,− Ok for k = 8(2)22 with x = −0.999, in red, and the intersections with the same plane of ψ(t, t∗ ) for diﬀerent values of t∗ on the Jacobi levels of P Ok , k = 10(4)22, in blue. Note that these four blue curves are quite close and similar. The possible places for the transfer are the intersections of a WPs,− Ok curve with the corresponding blue curve. They are marked in magenta. For each k shown here two possible places are obtained. This plot is done using y, y˙ as variables.

3.4.2 Escape and conﬁnement in the Sitnikov problem This is an example to study escape/capture on a given problem of Celestial Mechanics using a very simple model. Two massive bodies of equal mass are moving on the z = 0 plane on elliptic orbits of eccentricity e around the common centre of mass, located at (0, 0, 0), with semimajor axis a = 1, while a body of negligible mass moves along the z-axis. The standing equations are z¨ = −

(z 2

z , + r(t)2 /4)3/2

r(t) = 1 − e cos(E),

t = E − e sin(E),

(3.26)

where E denotes the eccentric anomaly of the primaries. For e = 0 the problem has one d.o.f. and, hence, it is integrable. As a ﬁrst order system we have z˙ = v, v˙ = z(z 2 + r(t)2 /4)−3/2 , with the obvious symmetries S1 : (z, v, t) ↔ (z, −v, −t), S2 : (z, v, t) ↔ (−z, v, −t), and S3 : (z, v, t) ↔ (−z, −v, t). We can introduce E as new time variable (denoting = d/dE) and introduce a Hamiltonian formulation: 1 H(z, E, v, J) = (1 − e cos(E)) v 2 − (z 2 + (1 − e cos(E))2 /4)−1/2 − J. 2 A suitable Poincar´e section for the representation of orbits is Σ = {z = 0}, using (v, E) as local coordinates. Thanks to the symmetry and to avoid strong deformations we shall use, instead, (ˆ v , E), where vˆ = |v|(1 − e cos(E))1/2 . If the inﬁnitesimal mass escapes to inﬁnity, the massive bodies move in S1 (eventually, after regularization of binary collisions using Levi–Civita variables).

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Chapter 3. Dynamical Properties of Hamiltonian Systems 2

1

-1

-2

-2

-1

1

2

Figure 3.12: Left: A representation of the Sitnikov model. Right: For e = 0 plots of the orbits in the (z, v) variables for values of H equal to −1.5, −1.0, −0.5 and 0.

One talks of a periodic orbit at inﬁnity. A celebrated Theorem by Moser states the following. Theorem 3.4.1. The problem has periodic orbits at both z plus and minus inﬁnity, with invariant manifolds (orbits going to or coming from inﬁnity parabolically). For e small enough the manifolds intersect Σ in curves diﬀeomorphic to circles. These curves have transversal intersection, implying the existence of heteroclinic orbits from +∞ to −∞ and vice-versa. As a consequence one has non-integrability, embedding of the shift with inﬁnitely many symbols, existence of oscillatory solutions, escape/capture domains, etc. The PO at ∞ is parabolic or, topologically, weakly hyperbolic. The linearized map around the PO is the identity. To study the vicinity of these orbits we introduce McGehee variables (q, p) deﬁned as z = 2/q 2 , z˙ = −p. Then the equations of motion become −3/2 , Ψ = (1 − e cos(E))/4. (3.27) q = Ψq 3 p, p = Ψq 4 1 + Ψ2 q 4 If e = 0 the invariant manifolds are given as p = ±q(1 + q 4 /16)−1/4 . We shall denote as W±u,s the intersections of unstable/stable manifolds of ±∞ with Σ. Due to S3 , W±u coincide and also W±s coincide, but W+s , W−u have v > 0, while W−s , W+u have v < 0. Due to S1 , W+u and W−s are symmetric with respect to E = 0. We look for a parametric representation of the manifolds of the PO as p(E, e, q) = bk (e, E)q k = ci,j,k ei sc(jE) q k , (3.28) k≥1

k≥1 j≥0 i≥0

where bk (e, E) are trigonometric polynomials in E with polynomial coeﬃcients in e, ci,j,k are rational coeﬃcients, and sc denotes sin or cos functions. Note that the problem can be reduced to obtain invariant manifolds of ﬁxed parabolic points of discrete maps (think about the intersection of the manifolds with E = 0). In this context McGehee proved that the invariant manifolds are

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analytic except, perhaps, at q = 0, see [31]. In fact, a result of Baldom`a and Haro [1] shows that, generically, the 1-dimensional manifolds of ﬁxed parabolic points are of some Gevrey class (see the part on invariant manifolds in Section 3.2.1). From (3.27) and (3.28) the invariance of the manifolds can be written as −3/2 dbk Ψq 4 1 + Ψ2 q 4 (e, E)q k + = bk (e, E)Ψkq k+2 bm (e, E)q m . dE k≥1

k≥1

m≥1

(3.29) Equating coeﬃcients of powers of q in (3.29) leads to the recurrence 2m+1

3 n−3 1 − e cos(E) 1 − e cos E −2 = b n (e, E) + kbk (e, E)bn−2−k (e, E), m 4 4 k=1 (3.30) where m = n/4 − 1, deﬁned only for n multiple of 4. To solve the recurrence in (3.30) we ﬁrst note that for the unstable manifold of +∞ we have b1 = 1. One has b1 = −1 for the stable manifold. For a given value of n we can split the function bn as ˜bn +¯bn , where ¯bn denotes the average and ˜bn the periodic part. Given bn (e, E) equal to some known function (computed from the previous coeﬃcients) allows us only to compute the periodic part ˜bn . The average ¯bn is computed previous to the solution of the equation for b (e, E), to have a n+3 zero average function when we integrate. An essential fact is that b2 = b3 = b4 = 0. One has also b6 = b7 = b10 = 0, but this is not so relevant. Now it is a simple task to implement the computation of the coeﬃcients to high order. Using high order is important, because this allows us to have a good representation for large values of q. A large q allows us to start the numerical integration, to obtain the intersection W+u of the manifold with z = 0, at a moderate value of z. For instance, using terms up to order n = 100 one checks that the representation is good (error of the order of 10−16 ) for q = 1/3. Then the numerical integration can be started at z = 2/q 2 = 18. Figure 3.13 shows some results for diﬀerent values of e, displaying W+u and s W− , and using the (ˆ v , E) variables as polar coordinates. Note that the use of (|v|, E) would give curves extremely elongated to the right for e close to 1. Concretely, if the eccentricity is equal to 1 − δ then the horizontal variable in the plots √ reaches values ≈ 2/ δ. The values of the splitting angle at E = 0 and E = π on the section Σ are shown as a function of e in Figure 3.14. Note the quite diﬀerent behaviour when e → 1. This gives evidence of the transversality for all values of e. Summarizing, the steps to obtain the manifolds W+u and W−s and, hence, the splitting angle, are the following: (i) introduce McGehee coordinates to pass from (3.26) to a formulation around the periodic orbits at inﬁnity, as given by (3.27); (ii) look for a suitable representation, as the one in (3.28), in which the manifold is expressed as function of a distance to inﬁnity (q) and a periodic time variable

208

Chapter 3. Dynamical Properties of Hamiltonian Systems

2

2

1

1

-1

-1

-2

-2 -2

-1

1

2

2

2

1

1

-1

-1

-2

-2 -2

-1

1

2

-2

-1

1

2

-2

-1

1

2

Figure 3.13: The manifolds W+u , in red, and W−s , in blue, for diﬀerent values of e. Top

left: for e = 0.1. Top right: for e = 0.5. Bottom left: for e = 0.9. Bottom right: for e = 0.999. In all cases we use (ˆ v , E) as polar coordinates. 3

2

1

0 -1

-0.5

0.5

1

Figure 3.14: The splitting angle of the manifolds W+u and W−s in Σ. For positive values

on the horizontal axis the splitting angle at E = 0 is shown as a function of e. For the negative ones, the splitting angle at E = π is shown as a function of −e.

3.4. Applications to Celestial Mechanics

209

(E); write the invariance condition (3.29) and derive the recurrences, as given in (3.30); (iii) analyze the properties of the recurrences (symmetries, powers of e in the coefﬁcients of the trigonometric polynomials, etc); design and implement routines to obtain the desired numerical coeﬃcients; and (iv) select a suitable value of q for the current maximal order of the expansion, evaluate (3.28) for a sample of values of E for every desired value of e, and carry out the numerical continuation until z = 0. It is important to stress that, for other similar problems (RTBP planar or spatial, general, etc), to decide if an observed body will be captured or will escape, it is enough to obtain the manifolds and decide the actual position with respect to them. The case of the planar RTBP with a comparison between theoretical predictions and numerical results can be found in [28] and the related slides (I).

3.4.3 Practical conﬁnement around triangular points As mentioned at the beginning of Section 3.4, the triangular libration points are linearly stable in the 3D RTBP if μ is small enough. But, what can be said about nonlinear stability? For the 2D case, nonlinear stability is proved for μ ∈ [0, μ1 ) except for the couple of values μ2 , μ3 . A possible approach is to reduce to the study of a symplectic 2D map and to apply Moser theorem. There is an exceptional value for which the twist condition is not satisﬁed, but can be recovered as a weak twist to higher order via normal forms. In the 3D case, in principle, there is no way to avoid diﬀusion. Hence, initial conditions as close as we like to L4,5 can go far away from that point. But normal forms, or averaging, lead to the already mentioned Nekhorosev estimates, showing that one needs an extremely large time if one starts close enough to the libration point as discussed in Subsection 3.3.5. But these results, concerning domains of practical stability in the 3D case, give at most small regions around the triangular points. On the other hand one has found the so-called Trojan (and Greek) asteroids, for the Sun-Jupiter system, far away from L4,5 , even with relatively high inclination. Hence, it seems that the domain of practical stability for long times is much larger than what is given by theoretical predictions. It would be nice to search for the conﬁning mechanisms. A side problem is why Trojan-like bodies are not found in the Earth-Moon case. Certainly the Sun is guilty for that, the orbits equivalent to L4,5 for the Earth-Moon system being unstable even in simple models of the Earth-Moon-Sun motion. But this does not exclude the possibility that stable orbits exist with moderate inclination. Here we present some results which can help to understand the main mechanisms, see [48]. For diﬀerent reasons, many computations are done with initial

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conditions on the ZVS using (z, α, ρ) as parameters for a ﬁxed μ, as follows: x = μ + (1 + ρ) cos(2πα),

y = (1 + ρ) sin(2πα),

z = z0 ≥ 0,

α ∈ (0, 1/2),

x˙ = y˙ = z˙ = 0.

(3.31)

As for μ = 0 one must be in 1 − 1 resonance, it is convenient to look, starting at the ZVS, for initial conditions at rest, in the synodical frame, in the moment that an elliptic orbit with semimajor axis equal to the unity passes through the apocentre in the sidereal frame. That is, for values of (z, R = 1 + ρ) related by 1/2 z = 4(1 + R2 )−2 − R2 ψ = 1 − 21 w +

3 2 25 w

−

1 3 28 w

−

25 4 213 w

+

or (3.32) 33 5 216 w

+ O(w6 ),

where w = z 2 , ψ = ψ(z) = R2 . This suggests that we make plots using the variables (α, γ = 1 + ρ − ψ(z), z). (3.33) It is clear that L5 corresponds to ρ = 0, α = 1/3, z = 0. By symmetry, similar results are obtained for L4 . Also, by symmetry, it is enough to look for z ≥ 0. For the limit case, μ = 0, one would have γ = 0. Some reasons to start at the ZVS are: (i) Most of the i.c. non-leading to escape are on 3D tori. Hence, we scan a set of positive measure in the full phase space (not ﬁxing the Jacobi constant C). (ii) The results obtained can be used as a seed to obtain the relevant objects involved in the practical conﬁnement, either starting at the ZVS or not. 0.1 1

0.05

0.5

-0.5 -1

1

-0.05 0.95

1

1.05

Figure 3.15: Example of a transition for μ = 0.0001, α = 0.05, z = 0.3. The two tori

(conﬁned in red, escaping in blue) have values of ρ which diﬀer in 10−10 . We show the projections on (x, y) of the Poincar´e section through z = 0. Left: a global view. Right: a magniﬁcation. The separating unstable 2D torus or invariant curve in the section belongs . Note that the points in red are partially hidden by the ones in blue. to WLu,s 3

3.4. Applications to Celestial Mechanics

211

First, we show some results concerning the quasi-boundary between escape and practical conﬁnement. Figures 3.15 and 3.16 display, for a small value μ = 10−4 of the mass parameter, two diﬀerent kinds of objects which appear on the quasiboundary. We should mention that the relevant objects have codimension 1 in the full phase space. In the present case they have dimension 5. Typically, they are W u,s of central objects of dimension 4. These objects can be the centre manifolds of ﬁxed points of centre×centre×saddle type or the centre manifolds of 1-parameter families of periodic orbits of centre×saddle type (the parameter being, e.g., the value of the Jacobi constant). But it is clear that these W u,s do not coincide: there is some splitting. This is the reason why they are named quasi-boundaries. We note, for instance, that in the upper left plot of Figure 3.16 beyond the blue curve commented on the caption, one can guess another invariant curve (in the Poincar´e section, a 2D torus in the phase space) on top of the plot. The separation between conﬁned and escaping orbits is close to a double heteroclinic connection between the lower curve in blue and the upper one in red. But the related branches of these two partially normally invariant curves do not match exactly. There is some tiny splitting between the branches. 0.8 1

0.5 0.4 0

-0.5 -1

-0.5

0.5

-0.4

-0.8

0.4

0.655

0.5

0.65

-0.5 0.645

1 -1 0

-0.8

-0.4

0.4

Figure 3.16: Similar to Figure 3.15 but starting at α = 0.4, z = 0.6. Now the separating

. Top: initial part of Poincar´e iterates with many iterates unstable 2D tori are not in WLu,s 3 in blue, giving evidence of the lower unstable 2D torus and points escaping from it (left), and the separating lower unstable invariant curve alone (right) projected on (x, y). Bottom: The same curve projected on (x, z) ˙ (left), and the related 2D separating unstable torus in a (x, y, z) projection (right).

In Figure 3.17 we display a general view of the boundary. See comments on the caption. Typically, the transitions have been detected after a maximum

212

Chapter 3. Dynamical Properties of Hamiltonian Systems

integration time equal to 106 × 2π (in special cases 10, 102 or 103 times larger) and with a resolution of 10−6 in ρ; see slides (D) for other values of μ.

Figure 3.17: A 3D view of the detected boundaries of practical stability starting at

the ZVS for μ = 10−4 , shown in the (α, γ, z) variables. The inner (resp., outer) part corresponds to γ < 0 (resp. γ = 0). Note the sharp change on the behaviour of the boundary which occurs between z = 0.4 and z = 0.5.

We can make a rough scan of the boundaries for diﬀerent values of μ, both for the planar and spatial RTBP. We say rough in the sense that, typically, the maximal time to look for escaping has been reduced to 105 × 2π time units and that the grid we scan uses Δρ = 10−4 , then Δα equal to 2×10−4 in the planar case (5 × 10−4 in the spatial one) and Δz = 5 × 10−3 in the spatial case. The results are shown in Figure 3.18. Note that the eﬀect of the resonances is less important in the spatial case. This is due to the fact that, for some values of μ, the resonances destroy stability in the planar case, but still a large set of initial conditions is stable in the spatial case. The change of the frequencies when z increases is responsible for the minima being shifted to larger values of μ. From now on we concentrate on a ﬁxed value μ = 0.0002. The reasons for this choice are the following: (i) μ being small, the boundaries are sharper; (ii) it should be also possible to obtain some information by means of perturbation theory; (iii) it is close to the Titan-Saturn mass ratio. This small value of μ, however, raises a problem: the escape is relatively slow and, hence, the integration time is large. The methodology used (for the L5 case) is as follows:

3.4. Applications to Celestial Mechanics

213

3D and 2D 1

0.5

0.01

0.02

0.03

0.04

Figure 3.18: Statistics as a function of μ starting at the ZVC (planar case, in red) and at the ZVS (spatial case, in blue). This is normalized to the maximum, which for the planar case occurs at μ = 0.0014 with 282757 points, and for the spatial case it occurs for μ = 0.0017 with 19014882 points. Note the sharp eﬀect of the resonances in the planar case. Similar patterns are found for the H´enon map and in many other examples, see [46]. In the spatial case the eﬀect of the resonances is milder and delayed. In both cases some stability subsists even for μ > μ1 .

(i) Deﬁne some escape criterion (e.g., the (x, y) projection of the orbit enters some wedge near the negative y-axis, or the orbit comes too close or too far from the primary, or too close to the secondary). (ii) Scan a set of initial conditions for short time (e.g., 104 × 2π, using some grid with small steps Δα, Δρ, δz). Look at every initial point on the grid, for ﬁxed z, as a pixel. Keep the pixels non leading to escape. (iii) Repeat for longer time (e.g., 5×104 ×2π) for the pixels at a distance (counted in the sup norm) less than d pixel units from the ones which already escaped (typically, we take d = 5). The tested points are marked depending on whether they escape or they remain. Iterate the scan until no more points have to be tested: all the ones at distance less than or equal to d from escaping points have been tested and remain. Repeat two more times for longer and longer integration time (25 × 104 × 2π, 106 × 2π). (iv) Eventually do additional reﬁnements of ρ for ﬁxed α, z. Figures 3.19 and 3.20 show some results for μ = 0.0002 displaying, for different values of z, the set of non-escaping points starting on the ZVS and the boundaries of the domain. See the captions for the variables used to represent the results. Note that the domain of practical stability contains, for the planar case z = 0, stable points quite close to the L3 (α ≈ 0). In the spatial case there are stable orbits which reach z as large as 0.865 and, as the value of ρ for these orbits reaches ≈ −0.181 they have a maximum inclination exceeding 46 degrees.

214

Chapter 3. Dynamical Properties of Hamiltonian Systems z=0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.825, 0.850, 0.865

-0.04

-0.08

-0.12

-0.16 0.5

0.4

0.3

0.2

0.1

Figure 3.19: For μ = 0.0002 the subsisting points, starting at the ZVS for 12 diﬀerent z values, given on the top of the plot. The coordinates used for the representation are (α, ρ). z=0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.825, 0.850, 0.865 0.008

0.004

-0.004

-0.008 0.5

0.4

0.3

0.2

0.1

Figure 3.20: Boundaries of the domains shown in Figure 3.19 using the paraboloid like corrections. That is, as vertical variable one has used γ, as deﬁned in (3.33) instead of ρ.

Some sections of the boundary for μ = 0.0002, for several values of α, are shown in Figure 3.21.

3.4. Applications to Celestial Mechanics

215 alpha=0.05, 0.12, 0.25, 0.33, 0.40, 0.435

alpha=0.05, 0.12, 0.25, 0.33, 0.40, 0.435 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

-0.16

-0.12

-0.08

-0.04

0.02

0 -0.008

-0.004

0.004

0.008

Figure 3.21: Some sections of the boundary starting at the ZVS for diﬀerent values of α. Left: in the (ρ, z) variables. Right: using the (γ, z) variables. The curves for α = 0.05, 0.12, 0.25 are plotted in red, and are easily seen on the right-hand plot going away from (0, 0) and each line encircling the previous one. The curve for α = 0.33 is displayed in blue. This is the largest one. Finally the curves for α = 0.40 and α = 0.435 are plotted in magenta. Last one does not reach z = 0. Still many things must be completed even for this small μ for which the boundaries tend to be rather sharp, because they are associated to relatively small splitting. The problem becomes more rough for the Sun-Jupiter case, because then one starts to see the eﬀect of some island-like structure. For the Earth-Moon case the behaviour is quite wild due to the strong eﬀect of resonances. The Earth-Moon mass ratio is not far from the 3:1 resonance value μ3 .

3.4.4 Inﬁnitely many choreographies in the three-body problem In the Newtonian N -body problem with all masses equal to 1, we can consider very simple solutions in the planar case, the like N -gon relative equilibrium solutions. Due to the homogeneity one can scale time and distance so that it is enough to consider solutions with period 2π. The N bodies move on a circle of radius R such that −1 −2 2R3 = ΣN . j=1 (2 sin(jπ/N )) It is clear that all the bodies move on the same path in the plane. Hence, the following is a natural question: are there other periodic solutions such that all bodies with equal masses move on the plane along the same path? At the end of the twentieth century a solution with 3 bodies on the same planar curve, diﬀerent form a circle, was proved to exist by Chenciner–Montgomery [6]. Also, Moore [33] found the same orbit in a previous numerical work in a diﬀerent context, a few years before. The path of this solution is the very popular ﬁgure eight curve and is displayed in Figure 3.22. Immediately, one can pose the question for N > 3 and for other shapes of the path. These solutions are called choreographies because of the dancing-like

216

Chapter 3. Dynamical Properties of Hamiltonian Systems

0.5

-0.5 -1

1

Figure 3.22: The ﬁgure eight solution of the three-body problem. The initial positions of the bodies are marked as black points. For concreteness, we can assume that at t = 0 the body located at the origin moves to the right, up. This forces the motion of the other two.

motion of the bodies seen in animations, see [5, 45]. More precisely, they should be called simple choreographies because they are on the same curve; we use the term k-choreographies for bodies moving on k diﬀerent curves. Slides (F) provide some examples and links to animations. One can also introduce the notion of relative choreographies if they are seen as choreographies in a uniformly rotating frame. Two choreographies which diﬀer only by a rotation, by scaling, change of orientation, symmetry, etc, should be seen as the same. Returning to simple choreographies in a ﬁxed frame (or absolute choreographies) what one tries to ﬁnd is some 2π–periodic function ψ : S1 → R2 such that if the body j is located at qj (t) = ψ(t − (j − 1)2π/N ) for j = 1, . . . , N , we have a solution to the equations of motion. Another natural question arises: are there other choreographies of the threebody problem diﬀerent from the ﬁgure eight? A simple observation is that at some t > 0, relatively small, the three bodies in Figure 3.22 will be in an isosceles conﬁguration. Such a conﬁguration is deﬁned, for instance, assuming that at some moment of time the bodies 2 and 3 have positions and velocities given by x3 = x2 ,

y3 = −y2 ,

x˙ 3 = −x˙ 2 ,

y˙ 3 = y˙ 2 .

(3.34)

The conditions for m1 are determined from the centre of mass integrals. This isosceles triangle has a symmetry axis passing through m1 . Assume that after some time τ the bodies pass through another isosceles conﬁguration, concerning positions, with the body m2 in the symmetry axis deﬁned by the positions of m3 and m1 , and that the velocities are close to satisfy the isosceles condition. Let β be the angle between the former symmetry axis (the xaxis) and the new one. A reﬁnement is done to satisfy the full isosceles conditions

3.4. Applications to Celestial Mechanics

217

with good accuracy (see the end of this subsection). Then, after rotating positions and velocities at τ by an angle −β, we have an isosceles conﬁguration with the same symmetries concerning velocities than the initial one. The only change is a circular permutation of the bodies with change of orientation. Then the action of the semi-direct product of Z2 and Z3 (symmetry and permutation of the bodies) produces a relative choreography with period T = 6τ and rotation 6β. If β is kπ, k ∈ Z, we have an absolute choreography, symmetric with respect to the xaxis. This has been applied to ≈ 109 initial conditions. Near 3 × 105 relative choreographies have been found and, by continuation of each one of them with respect to the angular momentum, many (345 up to now) absolute, non-equivalent, choreographies have been found. It is clear that several relative choreographies can lead, by continuation, to an absolute choreography equivalent to another one found previously, and these are not counted. It is checked that some of these new threebody choreographies seem to belong to families. An example is shown in Figure 3.23. See [44] for other families. Figure 3.23 suggests to try to continue the family for an increasing number of loops. Now the continuation has to be done with respect to integers and not in a continuous way. But using extrapolation of the data from the previous loops it has been possible to continue the family without any problem (using quadruple precision and high order extrapolation) until the solution shown in Figure 3.24. The natural conjecture is that there are inﬁnitely many choreographies in this family. There is an easy description of that solution. One of the bodies (say, the red one) moves close to an elongated ellipse while the other two (green and blue) move in a close binary, with its centre of mass close to an ellipse. When the three bodies approach the centre of mass there is an exchange: the blue body moves close to an elongated ellipse and the red and green form a binary in turn. At the end of this we have traveled 1/3 of the period. The bodies return to the initial position with a cyclic permutation RGB → GBR. One should stress that when they approach the centre of mass the bodies are not close to triple collision. Preliminary results seem to indicate that the minimal value of the moment of inertia along the orbit is strictly decreasing with the number of binary loops, tending to a positive constant. It should be mentioned that, among the 345 absolute choreographies available, one can identify several families. It is not excluded that some of these families contain inﬁnitely many elements. But it can also happen that a couple of families merge together in a saddle-node bifurcation. The steps for that application are as follows: (i) To obtain initial data in isosceles conﬁguration one can prescribe some negative energy. Then we give values of (x2 , y2 ) and determine the positions of the other masses. Because of the symmetries we can select x2 > 0, y2 < 0. A bound on the domain is obtained because the kinetic energy should be

218

Chapter 3. Dynamical Properties of Hamiltonian Systems l=12

l=8

l=4 0.3

0.3

0.3

-0.3

-0.3

-0.3

l=16

l=24

l=20

0.3

0.3

0.3

-0.3

-0.3

-0.3 -0.4

-0.4

0.4

l=28

0.4

-0.8

0.3

-0.3

-0.3 0

0.4

0.8

0.8

0.4

l=36

0.3

-0.4

-0.4

l=32

0.3

-0.8

0.4

-0.4

0.4

-0.4

0.4

-0.4

-0.3 -0.8

-0.4

0.4

0.8

-0.8

-0.4

0.4

0.8

Figure 3.23: Choreographies of the three-body problem belonging to a family. The paths of the three bodies during 1/3 of the period are shown in diﬀerent colors. The positions of the bodies in the initial isosceles conﬁguration and the ones after 1/6 of the period are also shown. To display the solutions with the same scale in x and y variables, the coordinates have been exchanged. Now, for these choreographies, the symmetry axis is the vertical one and for this family both isosceles conﬁguration (at t = 0 and after 1/6 of the period) are symmetrical the one from the other with respect to the horizontal axis. Counting the little inner loops (for instance, the ones in red) the number increases from 1 to 9 from top to bottom and from left to right. The value on top of each plot refers to the total number of small loops, either in red, blue or green. non-negative. The possible values of (x˙ 2 , y˙ 2 ) are parametrized by an angle γ ∈ [0, 2π]. (ii) Then, we proceed to the integration of (3.1) with the selected initial conditions, looking for a passage near another isosceles conﬁguration. A maximal time is used (e.g., 5 units) and the attempt is stopped if the bodies move too far or they become too close. If a candidate is obtained a reﬁnement is done by Newton’s method, to have a good approximation to an isosceles symmetry after 1/6 of the period. For the reﬁnement we ﬁx γ and leave (x2 , y2 ) as free variables to satisfy the isosceles condition for the velocities when it is satisﬁed by the positions. (iii) Next we carry out continuation by changing the angular momentum, looking

3.4. Applications to Celestial Mechanics

219

Figure 3.24: Top: a choreography of the three-body problem of the same family of the ones shown in Figure 3.23. In each of the binary portions the bodies in the binary make 200 revolutions around the centre of mass of the binary, while the third body moves close to an elongated ellipse. Only 1/3 of the orbit is shown. The remaining parts are obtained by cyclic permutations. Bottom: a magniﬁcation of the central part of the top.

for an absolute choreography. Continuation is stopped if the bodies approach a collision. The new absolute choreographies are stored in a list. If they are already in the list, they are discarded. Later, for our present goal, we select the ones which belong to the family as shown in Figure 3.23. (iv) Finally the family is continued with respect to the number of loops. An extrapolation based on the previously computed loops allows us to have a very good guess. Newton’s method converges in few iterations.

3.4.5 Evidences of diﬀusion related to the centre manifold of L3 In this last application we consider the 3D RTBP for a small value μ = 0.0002, like we used in Subsection 3.4.3. Our goal is to give evidence of the diﬀusion when we consider the unstable dynamics originated by the unstable/stable manifolds of the part WLc 3 ,C of the centre manifold WLc 3 of L3 , for a given value C of the Jacobi constant. For concreteness, we use the value C = 2.95998466228. To have a feeling of the meaning, let us say that for that value of C the vpoL in WLc 3 ,C has values of z going from −0.2 to +0.2. Beyond the vpoL , the WLc 3 ,C contains 2D tori, the hpoL , some tiny chaotic domains, and the additional periodic orbits related to these domains. Using the methods of Subsections 3.3.1 and 3.3.3, we can compute both periodic orbits and several tori. It is simpler to represent the tori as ICs of the Poincar´e map P associated to the section Σ := {z = 0, z˙ > 0}. In this application we shall use once and again Σ and P. As we ﬁxed also the value of C, we have to consider a discrete

220

Chapter 3. Dynamical Properties of Hamiltonian Systems

map in a 4D space that we denote as ΣC . The ICs are hyperbolic normally to the centre manifold. Hence, we can compute its manifolds, say WCu , WCs , for a given curve C. Note that these manifolds are 2D and to visualize them we can compute a section through some codimension-1 manifold in ΣC (e.g., an hyperplane Π). A suitably chosen Π gives as WCu ∩ Π a closed curve, say Cu . In a similar way we can obtain Cs . Of course, these two curves in ΣC ∩ Π, which is 3D, do not intersect generically, as opposite to WCu and WCs which are 2D in the 4D space ΣC , for which one expects to have intersections, but not necessarily located in Π. But we can have a feeling of their relative position by looking at Cu and Cs . Figure 3.25 illustrates what has been said. In the left plot several ICs are shown, as well as the point corresponding to the vpoL . Note that the largest IC is quite close to the hpoL . The 2D torus corresponding to this last IC has values of z which range in the small interval [−0.017, 0.017]. The hpoL , which is contained in z = 0, is located outside the largest IC shown at a distance ≈ 0.004. The right plot displays Cu and Cs for several ICs, using as Π the hyperplane deﬁned by √ y = − 3(x − μ). One detects, visually, that for tori close to the vpoL the curves are quite close. The diﬀerence increases going outside, away from the vpoL , and decreases again when approaching the hpoL . This will be one of the relevant facts to explain the results obtained. 0.4 0.2 0.2 0 0 -0.2 -0.2 0.7

0.8

0.9

1

-0.4

-0.2

0.2

-0.4 0.8

0.9

1

1.1

1.2

Figure 3.25: Left: invariant curves obtained as intersections with Σ of some tori in the

WLc 3 ,C for C = 2.95998466228 projected on the (x, y)-plane. The vpoL orbit for this value of C has z ∈ [−0.2, 0.2] and corresponds to the blue point. The√blue curve will be used in the computations reported here. Right: sections with y = − 3(x − μ) of the Poincar´e sections of the unstable (red) and stable (blue) manifolds of some of the tori. For the 3D view we use the (y, y, ˙ z) ˙ variables.

Figure 3.26 shows the projection in (x, y) of the ﬁrst 105 iterates under P starting at a point close to the blue curve, say Cb , in Figure 3.25, left. The ﬁrst iterates follow closely the upper part of WCub and return near Cb close to the upper part of WCsb (or of some other nearby curve). As it is well known, next iterates can continue going up or down, as happens after every return near Cb , in a quasirandom

3.4. Applications to Celestial Mechanics

221

way. For completeness, the manifolds of vpoL are also shown (displayed in blue). This behaviour suggests that, at the successive returns near WLc 3 ,C , the Poincar´e iterates can approach diﬀerent tori (2D in the phase space) on that centre manifold. That is, a typical mechanism of diﬀusion thanks to chains of heteroclinic connections of diﬀerent tori. But there are also tori (3D for the Hamiltonian ﬂow, 2D for P) close to these manifolds. Among these tori one ﬁnds the ones close to the boundary of the practical stability domain for L5 , as seen in Subsection 3.4.3. Looking at Figure 3.19 one checks that they reach values of α very close to 0 (the value of α for L3 ) up to z = 0.4. The successive points can remain for a large number of iterations, say 106 and even 108 in some tests, close to one of these tori, to one of the tori in the symmetric domain around L4 , or even tori which visit a vicinity of both L5 and L4 (with an (x, y) projection of the iterates in Σ similar to the red points in Figure 3.26). The tori are very sticky, see Subsection 3.3.5. As a consequence, the orbit of a point should consist of passages from the vicinity of the W u of one of the ICs to the vicinity of the W s of another IC (or, perhaps, the same one) with long stays near tori of one of the three types described. 1

-1 -1

1

Figure 3.26: Starting at a point very close to the invariant curve in blue in Figure 3.25

we have computed the ﬁrst 105 intersections with Σ. The plot shows the projections on the (x, y)-plane. As a reference, we also show in blue the initial part of the manifolds of the vpoL . The lack of coincidence of these last manifolds is not seen with the present resolution.

To have evidence of this expected behaviour, we have taken 1920 points close to Cb (the blue curve in Figure 3.25 left). For every initial point we record the ﬁrst 5×106 Poincar´e iterates, except if some kind of escape is detected. A typical escape occurs when, going the iterates to the left, either near the upper or lower part of Figure 3.26, they approach the location of the secondary. After this encounter, the successive iterates can move close to the primary, escape far away or even return several times near the secondary. Anyway, only for 37 of the 1920 initial conditions escape was detected. Certainly the initial conditions will lead to escape if the number of Poincar´e iterates is largely increased, at least on this level of the

222

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Jacobi constant. See later for some tests with initial data taken near the vpoL . To visualize the diﬀusion and to display a moderate amount of data we have computed passages of the Poincar´e iterates through a narrow slice around x = 0. Only from time to time an iterate falls in the slice. For instance, among the 1920 × 5 × 106 computed Poincar´e iterates (and except for the few iterates lost because of escape) only ≈ 3.2 × 106 fall in the slice |x| < 10−3 . The passage can occur in the upper part going from right to left (inner transition) or from left to right (outer transition), and also from right to left (outer transition) or from left to right (inner transition) in the lower part (see Figure 3.26). The variables used in Σ are (x, y, x, ˙ y). ˙ Due to the symmetries, the inner upper and inner lower transitions are symmetrical, with the changes (x, y, x, ˙ y) ˙ ↔ (x, −y, −x, ˙ y), ˙ and the same occurs for the outer ones. Using only the points falling into the slice up to a maximum of 105 iterates for all the initial conditions, the results (inner and outer upper transitions) are shown in Figure 3.27 left. The blue points, P− to the left and P+ to the right, correspond to the intersections with x = 0 of the manifolds of the vpoL . The point u P− is the ﬁrst intersection of Wvpo with x = 0, and P+ is the ﬁrst intersection of L s Wvpo with x = 0. The y coordinate of P− is smaller than the one of P+ . In both L cases we refer to the manifolds of vpoL as seen in Σ. Compare with the section through x = 0 of the upper part of the blue curves in Figure 3.26. Note also that in Figure 3.27 we display y − 1 as horizontal coordinate, while y˙ is used for the vertical one. To see the behaviour when the number of iterates increases, the right part of Figure 3.27 shows the evolution when we consider iterates in the slice after a maximal number of iterations going from 105 to 8 × 105 and, later, to 5 × 106 (from green to blue and then to red). The points are plotted in the reverse order. So, blue points hidden red ones and green points hidden blue ones. In magenta we show the location of P− . To prevent from too heavy ﬁles we take the narrower slice |x| < 10−4 and only show iterates when moving in the upper part to the left, that is, upper inner transitions. It is interesting to display statistics of the process. A simple measure is the evolution of the distance of the iterates to the point P− , marked in magenta in Figure 3.27 right. We use the slice |x| < 10−3 and all the Poincar´e iterates (up to 5 × 106 for the 1920 initial points, except for 37 points which escape, after escape is detected). Then we compute the distances rk,i to P− in the (y, y) ˙ variables, where i is the index of the initial point and k the number of the Poincar´e iterate. One takes samples of the rk,i for all the indices i and for ranges of k of the form ((j − 1)M, jM ], j = 1, . . . , 100, with M = 50, 000. The samples can be labelled by the ﬁnal value of k. The Figure 3.28 displays, on the left, the behaviour of the average distance as a function of the ﬁnal value of k in the range of values of k in the sample, while the behaviour of the standard deviation is shown on the right. For these computations both inner transitions (upper and lower) have been taken into account, in order to have larger samples (the total number of inner transitions amounts to 1643007).

3.4. Applications to Celestial Mechanics 0.1

223 0.2

0.1

-0.1

-0.1 -0.1

0.1

-0.2 -0.2

-0.1

0.1

Figure 3.27: For the set of points described in the text we show the (y − 1, y) ˙ projections

using diﬀerent slices and times, for y > 0 (for y < 0 it is similar). Left: the slice is deﬁned as |x| < 10−3 and we restrict to the ﬁrst 105 Poincar´e iterates of the initial points. In green (resp., red) the points when the iterates move to the left (resp., to the right) when looking at them projected on (x, y). We also show the location of P± , as described in the text. Right: points in the upper inner transitions. In red (resp., blue, green) we plot the points on the slice for a number of Poincar´e iterates up to 5×106 (resp., up to 8×105 , up to 105 ).

The results deserve some discussion. We can consider a diﬀusion process but, as the rate of diﬀusion is related to the passage from some 2D torus (invariant curve in the Poincar´e section) to a nearby one, from the comments preceding Figure 3.25, the rate of diﬀusion is not constant. It increases going away from the vpoL and then it decreases again when approaching the hpoL . From the left plot in Figure 3.28 it seems that the average is still in a range where the diﬀusion rate is increasing. This asymmetry is what produces the increase of the average. Note that the value of the distance to P− for the ﬁrst iterates which fall in the slice has an average of ≈ 0.0597. Concerning the standard deviation, one should mention that it takes a not so small value (≈ 0.005) for k = 50, 000 (the ﬁrst displayed point). One of the reasons for this is that, looking at the green points in Figure 3.27, one checks that they are scattered around an ellipse, not a circle. Also, after 50,000 iterates the scattering is non-negligible. One can mention that a good ﬁt of the data for the standard deviation, as a function of the number of Poincar´e iterates, k, is of the form σ ≈ c(a0 + a1 k + a2 k 2 )1/2 with a0 , a1 > 0, a2 < 0, and c a small positive constant. The negative character of a2 should be due to the decrease of the diﬀusion rate when going to the outer curves in Figure 3.25. Furthermore, when the distance d to P− reaches a value d∗ less than, but not too far from 0.18, the orbits quickly escape. One can check that the upper part of the unstable manifold of the hpoL has a ﬁrst intersection with x = 0 on a curve, similar to a circle, for which the distance to P− takes an average value equal to 0.2. Hence, we can consider this as a diﬀusion process with varying diﬀusion rate (ﬁrst increasing, later decreasing, as a function of the distance to P− ) and with

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Chapter 3. Dynamical Properties of Hamiltonian Systems

0.025 0.075 0.02 0.07 0.015 0.065 0.01

0.06 0.005 0

1

2

3

4

5

1

2

3

4

5

Figure 3.28: The left (resp., right) plot shows (in red) the evolution of the average (resp., standard deviation) for ranges of k of the form ((j − 1)M, jM ], j = 1, . . . , 100, with M = 50, 000. The horizontal variable in the plots refers to millions of Poincar´e iterates. For comparison, the blue lines show the same results, with a reduced set of initial points, for computations done using quadruple precision. See the text for details. an absorbing barrier: reaching d = d∗ the points disappear from the system. It is worth commenting also that, as an additional check, preliminary computations concerning diﬀusion and the related statistics have been carried out using quadruple precision. The size of the sample of initial points has been reduced by a factor of 4. The number of escapes before reaching N = 5 × 106 is 9, in good agreement with the previous result. Note that now the samples for the statistics are smaller, which gives slightly larger errors in the determination of average and standard deviation. For comparison, the results are displayed in blue also in Figure 3.28. Concerning escape, the following experiment has been carried out. A total of 625 initial conditions has been taken in Σ at distances of the order of 10−13 from the intersection of the vpoL with Σ. Poincar´e iterates have been computed up to a maximum of 109 . The ﬁrst escape is produced after a number of iterates close to 65 × 105. Only 13 points subsist for the full 109 iterates, most of them spending a big part of the iterations very close to invariant tori. This is, again, related to the stickiness of these tori. A plot of the number of points which subsist after k iterations, for values of k multiples of 107 is shown in Figure 3.29. Furthermore, taking initial data close to the 9 outermost tori in Figure 3.25 (again using samples of 625 points), one checks that all points escape, and that the average number of iterates for the escape decreases in an exponential way when we approach the outer torus. If the same experiment is done with 625 initial points close to the hpoL , the result is that all of them escape. In that case, as the orbit lives in z = 0, one can count the number of crossings of the orbits through the section x = 0, either with y > 0 or with y < 0, and either with x˙ > 0 or with x˙ < 0. The average number of such crossings is 14175. Note that, in contrast with the passage of Poincar´e iterates through a slice around x = 0, it happens that

3.4. Applications to Celestial Mechanics

225

there are outer and inner, upper and lower crossings both with x˙ > 0 and with x˙ < 0. See [48] for an explanation of this fact. These results, displayed in Figure 3.29, require a few comments. Up to 64.9 million iterates there is no escape. Only 14 points escape before 108 iterates. Then, up to ≈ 3 × 108 iterates the number of subsisting points is nearly linear in the number of iterates, that is, a rate of decrease close to a constant. Finally, up to ≈ 9 × 108 the rate of escape is slightly below an exponential one. The last escape was produced around 870 million iterates. To explain these changes is a nice open problem.

600

400

200

0 0

200

400

600

800

1000

Figure 3.29: Statistics of the number of non-escaping points, starting close to the vpoL , as a function of the number of Poincar´e iterations. For the simulations one has used a sample of 625 initial points. In the horizontal axis the number of iterations is shown in millions.

A basic ingredient for this application is to have an eﬃcient method to compute Poincar´e iterates. The steps are the following: (i) The computation, stability properties and unstable direction of the vpoL , as ﬁxed point of the Poincar´e map, is an easy task. The invariant curves of P are computed by looking at a representation of the variables (x, y, x, ˙ y) ˙ as Fourier series in a parametrization angle, using a number of harmonics between 6 and 26, depending on the torus, as explained in Subsection 3.3.3. The symmetries imply that, setting the origin of the angle at the minimal value of x, both x and y˙ are even, while y and x˙ are odd. In this way, the left plot in Figure 3.25 has been obtained. As a side comment we remark that the rotation numbers are of the order of 10−4 and decreasing when going away from the vpoL . This produces some problems in the condition number of the linear systems to be solved in the Newton iterations. (ii) The next step is the computation of invariant unstable/stable manifolds of the invariant curves. The reversibility implies that it is enough to compute the unstable ones, the stable being then recovered by the symmetries.

226

Chapter 3. Dynamical Properties of Hamiltonian Systems We recall that the manifolds have a parametrization as a function of an angle and a distance to the curve. A fundamental domain is diﬀeomorphic to a cylinder. Looking for points such that after some number of iterations are on an hyperplane Π requires a continuation method (e.g., to have the starting distance as a function of the angle) or any similar device. This has been used for the right-hand plot in Figure 3.25. The plots in Figure 3.26 follow immediately from the computation of Poincar´e iterates.

(iii) To produce Figure 3.27 only requires the computation of Poincar´e iterates, detection of the passage through a given slice and whether an inner or outer, upper or lower passage occurs. These are elementary tasks, despite the computational cost being high. The statistics can be produced by elementary means. Note that the diﬃculties mentioned in item (i), about the smallness of the rotation number, could be expected a priori. The problem in this region of the phase space is a tiny perturbation of the two-body problem in synodical coordinates. If μ → 0 the limit is the two-body problem, without the singularities which occur in the case of L1 and L2 due to the presence of the secondary, which lead, under suitable scaling, to a limit non-integrable case which is Hill’s problem, see [35, 49]. Hence, for μ → 0, the rotation numbers of the ICs like the ones in Figure 3.25 tend to zero. Concretely they are O(μ), in contrast with the hyperbolicity at L3 and √ also at the ICs, the vpoL and the hpoL , which is O( μ). The possible resonances are of a so high order that they become undetectable. The diﬀusion comes only from the eﬀect of the heteroclinic connections of the manifolds of these ICs. The situation is more complex if there is also a relevant amount of hyperbolicity in the centre manifold itself. See related topics in [13]. Summarizing: one has good evidence of the existence of diﬀusion associated to the centre manifold of L3 on levels of the Jacobi constant not too far from the value at that point. Certainly one can produce escape, due to the eﬀect of the secondary and even for μ as small as 0.0002, but the escape time is large. Anyway, there are many topics which require further research.

Acknowledgments Some of the topics presented in this Chapter have been partially supported by grants MTM2010-16425 and MTM2013-41168-P (Spain), 2009 SGR 67 and 2014 SGR 1145 (Catalonia). The author is grateful to the Centre de Recerca Matem` atica where he was staying during the preparation of the written version of these lectures.

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