Applied Mathematical Sciences Volume 138

Editors J.E. Marsden L. Sirovich

Advisors s. Antman J.K. Hale P. Holmes T. Kambe J. Keller K. Kirchgassner BJ. Matkowsky C.S. Peskin

Springer Science+Business Media, LLC

Applied Mathematical Sciences

1. J ohn: Partial Differential Equations, 4th ed. 34. Kevorkian/Cole: Perturbation Methods in Applied 2. Sirovich: Techniques of Asyrnptotic Analysis. Mathematics. 3. Hale: Theory of Functional Differential Equations, 35. Carr: Applications of Centre Manifold Theory.

2nd ed. 36. Bengtsson/Ghii/Klillen: Dynamic Meteorology: 4. Percus: Combinatorial Methods. Data Assimilation Methods. 5. von MiseslFriedrichs: Flnid Dynamics. 37. Saperstone: Semidynamical Systems in Infinite 6. FreibergerlGrenander: A Short Course in Dimensional Spaces.

Computational Probability and Statistics. 38. Lichrenberg/Lieberman: Regular and Chaotic 7. Pipkin: Lectures on Viscoelasticity Theory. Dynamics, 2nd ed. 8. Giacoglia: Perturbation Methods in Non-linear 39. Piccini/StampacchiaIVidossich: Ordinar)'

Systems. Differential Equations in Rn.

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Equations. 43. Ockendon/l'aylor: Inviscid Fluid Flows. 14. Yoshizawa: Stability Theory and the Existence of 44. Pazy: Semigroups of Linear Operators and

Periodic Solution and Almost Periodic Solutions. Applications to Partial Differential Equations. 15. Braun: Differential Equations and Their 45. GlashofflGustajson: Linear Operations and

Applications, 3rd ed. Approximation: An Introduction to the Theoretical 16. Lejschetz: Applications of Algebraic Topology. Analysis and Numerica! Treatrnent of Semi-17. CollatzIWetter/ing: Optimization Problems. Infinite Programs. 18. Grenander: Pattern Synthesis: Lectures in Pattern 46. Wilcox: Scattering Theory for Diffraction

Theory, VoI. 1. Gratings. 19. Marsden/McCracken: Hopf Bifurcation and Its 47. Hale et al: An Introduction to Infinite Dimensional

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Equations. 49. Ladyzhenskaya: The Boundary-Value Problems of 21. CourantlFriedrichs: Supersonic Flow and Shock Mathematical Physics.

Waves. 50. Wilcox: Sound Propagation in Stratified Fluids. 22. RouchelHabets/Laloy: Stability Theory by 51. GolubitskylSchaeffer: Bifurcation and Groups in

Uapunov's Direct Method. Bifurcation Theory, VoI. 1. 23. Lamperti: Stochastic Processes: A Survey of the 52. Chipot: Variational Inequalities and Flow in

Mathematical Theory. Porous Media. 24. Grenander: Pattern Analysis: Lectures in Pattern 53. Majda: Compressible Fluid Flow and System of

Theory, VoI. n. Conservation Laws in Several Space Variables. 25. Davies: Integral Transforms and Their 54. Wasow: Linear Turning Point Theory.

Applications, 2nd ed. 55. Yosida: Operational Calculus: A Theory of 26. KushnerlClark: Stochastic Approximation Hyperfunctions.

Methods for Constrained and Unconstrained 56. ChanglHowes: Nonlinear Singular Perturbation Systems. Phenomena: Theory and Applications.

27. de Boor: A Practical Guide to Splines. 57. Reinhardt: Analysis of Approximation Methods 28. Keilson: Markov Chain Models-Rarity and for Differential and Integral Equations.

Exponentiality. 58. DwoyerlHussainWoigt (eds): Theoretical 29. de Veubeke: A Course in Elasticity. Approaches to Turbulence. 30. Shiatycki: Geometric Quantization and Quantum 59. SanderslVerhulst: Averaging Methods in

Mechanics. Nonlinear DynamicaI Systems. 31. Reid: Sturmian Theory for Ordinary Differential 60. GhiVChildress: Topics in Geophysical Dynamics:

Equations. Atrnospheric Dynamics, DynarDO Theory and 32. MeislMarkowitz: Numerical Solution of Partial Climate Dynamics.

Differential Equations. 33. Grenander: Regular Structures: Lectures in

Pattern Theory, VoI. m. (continued jollowing index)

G. Haller

Chaos N ear Resonance

With 155 Illustrations

i Springer

G. Haller Division of Applied Mathematics Brown University Providence, RI 02912 USA haller@cfm.brown.edu

Editors

J.E. Marsden Control and Dynarnical Systems, 107-81 California Institute of Technology Pasadena, CA 91125 USA

L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA

Cover illustration: Invariant manifolds near resonance in a modal truncation of the forced nonIinear Schriidinger equation.

Mathematics Subject Classification (1991): 34CXX, 53-XX, 58FXX

Library of Congress Cataloging-in-PubIicatioJ1 Data HalIer, Gyiirgy.

Chaos near resonance/Gyiirgy Haller. p. cm. - (AppIied mathematical sciences; 138)

IncIudes bibIiographicaI references and index. ISBN 978-1-4612-7172-7 ISBN 978-1-4612-1508-0 (eBook) DOI 10.1007/978-1-4612-1508-0 1. Chaotic behavior in systems. 2. Resonance-Mathematics.

1. Title. II. Series. QI72.5.C45H345 1999 003'.857-DC21 98-53843

Printed on acid-free paper.

© 1999 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1999 Softcover reprint of the hardcover 1 st edition 1999

Ali rights reserved. This work may not be translated or copied in who1e or in part without the written permission of the pubIisher, Springer Science+Business Media, LLC except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive narnes, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such narnes, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

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987 654 3 2 1

ISBN 978-1-4612-7172-7

To my parents

Preface

Resonances are regions in the phase space of a dynamical system in which the fre­quencies of some angular variables become nearly commensurate. Such regions have a profound effect on the dynamics of the system, since they are rich sources of highly complex motions. In molecular dynamics, resonances are known to give rise to chaotic patterns, multiple time scales, and apparent irreversibility in the transfer of energy between different oscillatory states of molecules. In engineering struc­tures, interactions among resonant modes are responsible for most complicated dynamical phenomena, which again include energy transfer, multi-time-scale be­havior, and chaotic motions. It is of great practical importance to understand the common mechanism behind these irregular features, both qualitatively and quan­titatively. The circle of further applications ranges from nonlinear optics through celestial and fluid mechanics to electromagnetism.

The Theme of This Book

This book presents a unified approach to multi-time-scale chaotic dynamics near resonances. The theory developed here has been primarily motivated by experi­mental and numerical observations of complex behavior in resonant systems. The main theme is the detection of invariant structures in the phase space with a strong impact on the general evolution of the system. Such structures are slow or partially slow manifolds, fast multipulse connections among them, and slow-fast chaotic sets that coexist with, and are influenced by, the former objects. These structures will be used to establish the existence of chaotic jumping in different classes of

viii Preface

problems, ranging from two degrees of freedom to infinitely many. A number of applications will be discussed as motivational examples as well as test beds for the analytic methods.

A common feature of most modal equations arising in resonance studies is that they are near-integrable for small values of forcing and dissipation. In spite of this fact, analytic global perturbation studies of such problems rarely appear in the literature. The main reason is that the best-known global perturbation method, the Melnikov method, does not unravel their chaotic behavior for linear damping and sinusoidal forcing, the most frequent ingredients of simple models. More specifically, invariant tori or periodic solutions of the integrable limit are usually destroyed by such perturbations, and hence, calculations for the intersection for their stable and unstable manifolds do not make sense. One is therefore forced to look for other mechanisms that create complicated dynamics. Such mechanisms turn out to develop in regions where the classical Melnikov method is inapplicable, that is, near slow-fast invariant sets that become singular in the integrable limit. In this sense, this book aims to extend the current tool kit of dynamical systems to cover a large class of applications that have not been amenable to geometric global perturbation methods. The invariant sets one obtains in this fashion are more complicated and compare very well with experimental and numerical observations.

Why Would Somebody Read All This?

This monograph is written for the reader who does not wish to be (re)introduced to dynamical systems by yet another book, but rather, would like to see it fulfill its long-standing promise to solve real-life problems. I share the view that simple models are essential first steps in tackling any physical problem, but I also believe that more elaborate models often lead to a better understanding of the phenomenon. The multi-time-scale phenomena analyzed in this book require at least/our dimen­sions in modeling. In four and higher dimensions, we lose our usual "Euclidean" intuition and cannot prove theorems just by drawing simple pictures. For this rea­son, the use of tools from differential geometry, higher- or infinite-dimensional dynamical systems theory, or even functional analysis will often be unavoidable. Nevertheless, the beauty of the subject is that simple geometric intuition prevails and guides one amid complicated-looking formulas.

I believe that the topics covered in this book are of interest to both the theoretical dynamicist and the applied scientist. The former will find new results on global perturbation methods, homoclinic bifurcations, the geometry of energy transfer, diffusion in phase space, analysis of the addition of the "tail" to a normal form, generalized Silnikov orbits, and infinite-dimensional invariant manifolds, just to name a few. The latter will find problems from beam dynamics, structural mechan­ics, surface wave oscillations, turbulent boundary layer models, particle motion in the atmosphere, subharmonic generation in optical cavities, pUlse-propagation in optical fibers, and multi-degree-of-freedom rigid body dynamics.

Preface ix

The Contents

Chapter 1 is included for the convenience of those seeking a deeper understanding of the arguments behind the main theorems. It contains an introduction to some basic and some nonstandard topics in dynamical systems. Even though details will often be omitted, the reader will always find references for further reading. Chapter 1, as well as other chapters, builds on Appendixes A and B, which supply supplementary material from geometry and analysis.

Chapter 2 develops the underlying theory of homoclinic jumping and slow-fast chaos for finite-dimensional dynamical systems. This chapter is self-contained, giving all the details and proofs for both the dissipative and the Hamiltonian cases. However, the reader less concerned with details may skip the technical sections to find an informal discussion of the proofs of the main results. After the main theorems on homoclinic jumping, important special cases are described. This is followed by the discussion of slow-fast chaotic behavior in the dissipative and the Hamiltonian contexts, and the chapter ends with an extension to partially slow manifolds of higher codimension. Throughout the chapter, finite-dimensional ap­proximations of the nonlinear SchrOdinger equation will be used to illustrate the assumptions and the power of the theorems.

Chapter 3 is written primarily for those looking for sample applications in order to analyze their own problem. While this chapter shows the techniques of Chapter 2 "in action," it can be read without a thorough understanding of the preliminaries. It also flashes several classical tricks that make resonance problems more tractable. At the same time, it is more concerned than usual with relating averaged and truncated equations to their original counterparts.

Chapter 4 has a special emphasis on resonances in Hamiltonian mechanics. The topics covered here include resonant equilibria in three-degree-of-freedom systems, diffusion across resonances in several degrees of freedom, and near­resonance dynamics in a priori unstable systems. The applications include a model of the classical water molecule and concrete systems of coupled rigid bodies.

Chapter 5 contains an extension of Chapter 2 to evolution equations. This chapter is somewhat more technical than the rest and uses some elementary functional analysis from Appendix B. The theory is not complete due to difficulties arising from infinite-dimensionality. Nevertheless, the results are strong enough to explain complicated behavior in perturbed nonlinear Schrodinger equations and in coupled systems of such equations.

Acknow ledgments

I would like to thank Steve Wiggins and Jerry Marsden for their initial encour­agement and support. Several people have contributed to this book through their explanations, comments, and insights. For that I am grateful to Peter Bates, Carmen Chicone, Zaichun Feng, Ildar Gabitov, Yan Guo, Phil Holmes, Chris Jones, Edgar

x Preface

Knobloch, Gregor Kovacic, Martin Krupa, David Levennore, Jerry Marsden, Igor Mezic, Sri Namachchivaya, Ali Nayfeh, Drew Poje, Vered Rom-Kedar, Vassilios Rothos, Bjorn Sandstede, Larry Sirovich, Makoto Umeki, Ferdinand Verhulst, and Chongchun Zeng. I am indebted to Dave McLaughlin and lalal Shatah for their interest and mathematical help, and last but not least, for reminding me to divide by 2Jr.

The assistance of Govind Menon was invaluable in the preparation of this book. He read the whole manuscript in detail, caught many errors, and suggested several changes. I have also incorporated a number of valuable suggestions from Amadeu Delshams and Oliver O'Reilly, who kindly read and commented on several chap­ters. Yan Guo and Drew Poje also read parts of the manuscript and supplied useful comments.

I am grateful to the National Science Foundation and the Alfred P. Sloan Foundation for supporting my research that led to the birth of this book.

Finally, I greatly benefited from the hospitality of the Courant Institute, the Department of Mathematics at the University of Maryland at College Park, and the Department of Applied Mechanics at the Technical University of Budapest during the preparation of the manuscript.

Providence, Rhode Island G. HALLER

Contents

Preface vii

1 Concepts From Dynamical Systems 1 1.1 Flows, Maps, and Dynamical Systems ......... 1 1.2 Ordinary Differential Equations as Dynamical Systems 3 1.3 Liouville's Theorem . . . . . . . . 5 1.4 Structural Stability and Bifurcation 6 1.5 Hamiltonian Systems · ...... 7 1.6 Poincare-Cartan Integral Invariant 10 1.7 Generating Functions · ...... 11 1.8 Infinite-Dimensional Hamiltonian Systems 12 1.9 Symplectic Reduction · ...... 13 1.10 Integrable Systems. . . . . . . . . 15 1.11 KAM Theory and Whiskered Tori . 16 1.12 Invariant Manifolds · ..... 19 1.13 Stable and Unstable Manifolds .. 21 1.14 Stable and Unstable Foliations .. 24 1.15 Strong Stable and Unstable Manifolds 26 1.16 Weak Hyperbolicity · ......... 27 1.17 Homoclinic Orbits and Homoclinic Manifolds 28 1.18 Singular Perturbations and Slow Manifolds . 29 1.19 Exchange Lemma .......... 31 1.20 Exchange Lemma and Observability 33 1.21 Normal Forms ... 34 1.22 Averaging Methods · ........ 36

xii Contents

1.23 Lambda Lemma and the Homoclinic Tangle 40 1.24 Smale Horseshoes and Symbolic Dynamics. 41 1.25 Chaos .................... 45 1.26 Hyperbolic Sets, Transient Chaos, and Strange Attractors 47 1.27 Melnikov Methods. 48 1.28 Silnikov Orbits . . . . . . . . . . . . . . . . . . . . . . . 51

2 Chaotic Jumping Near Resonances: Finite-Dimensional Systems 56 2.1 Resonances and Slow Manifolds 56

2.1.1 The Main Examples . . . . . 57 2.2 Assumptions and Definitions .., . 58

2.2.1 An Important Class of ODEs 58 2.2.2 N -Chains of Homoclinic Orbits 62 2.2.3 Partially Slow Manifolds . . 65 2.2.4 N -Pulse Homoclinic Orbits . 66

2.3 Passage Lemmas . . . . . . . . . . . 68 2.3.1 Fenichel Normal Form . . . 68 2.3.2 Entry Conditions and Passage Time 72 2.3.3 Local Estimates 75

2.4 Tracking Lemmas . . . . 86 2.4.1 The Local Map . 87 2.4.2 The Global Map . 88 2.4.3 A Note on the Purely Hamiltonian Case 90

2.5 Energy Lemmas . . . . . . . . 91 2.5.1 Energy as a Coordinate . . 91 2.5.2 Energy of Entry Points . . 92 2.5.3 Improved Local Estimates 95 2.5.4 Energy of Projected Entry Points 96

2.6 Existence of Multipulse Orbits 98 2.6.1 Main Ideas. . . . . . . . . . . . 99 2.6.2 Existence Theorem . . . . . . . 103 2.6.3 Remarks on Applications of the Main Theorem 108 2.6.4 The Most Frequent Case: Chain-Independent

Energy Functions . . . . . . . . . . . . . . . . 110 2.6.5 Formulation With Other Invariants . . . . . . . III

2.7 Disintegration ofInvariant Manifolds Through Jumping 114 2.8 Dissipative Chaos: Generalized Silnikov Orbits. 116 2.9 Hamiltonian Chaos: Homoclinic Tangles . . . . 123

2.9.1 Orbits Homoclinic to Invariant Spheres. 125 2.9.2 The Case of n = 0: Orbits Heteroclinic

to Slow m-Tori ............ . 2.9.3 The Case of n = 0, m = 1: Orbits Homoclinic to Slow

2.9.4 2.9.5

Periodic Solutions ...... . Resonant Energy Functions . . Phase Shifts of Opposite Sign.

126

!27 131 135

Contents xiii

2.10 Universal Homoclinic Bifurcations in Hamiltonian Applications. . . . . . . . . . .

2.11 Heteroclinic Jumping Between Slow Manifolds 2.11.1 Partially Broken Heteroclinic Structures 2.11.2 Cat's Eyes Heteroclinic Structures . . .

2.12 Partially Slow Manifolds of Higher Codimension 2.12.1 Setup ..... . 2.12.2 Passage Lemmas 2.12.3 Tracking Lemmas 2.12.4 Energy Lemmas. 2.12.5 Existence Theorem for Multipulse Orbits. 2.12.6 Multipulse Silnikov Manifolds

2.13 Bibliographical Notes . . . . . . . . . . .

3 Chaos Due to Resonances in Physical Systems 3.1 Oscillations of a Parametrically Forced Beam

3.1.1 The Mechanical Model ..

136 141 142 145 147 147 151 151 152 152 155 157

159 160 160

3.1.2 The Modal Approximation . . . . . . 162 3.1.3 The Integrable Limit ......... 163 3.1.4 Homoclinic Bifurcations in the Purely Forced

Modal Equations ................ 166 3.1.5 Structurally Stable Heteroclinic Connections for the

Forced-Damped Beam . . . . . . . . . . . . . . .. 167 3.1.6 Chaos: Generalized Silnikov Orbits and Cycles for the

Forced-Damped Beam . . . 172 3.1.7 Numerical Study ............ 176

3.2 Resonant Surface-Wave Interactions ...... 178 3.2.1 Derivation of the Amplitude Equations . 179 3.2.2 The E = 0 Limit. . . . . . . 186 3.2.3 Chaotic Dynamics for E > 0:

Generalized Silnikov Cycles 188 3.2.4 Passage to the Limit E = fo 191 3.2.5 The Inclusion of the Oell V)

Time-Dependent Terms . . . 194 3.2.6 Comparison With the Simonelli-Gollub Experiment. 194

3.3 Chaotic Pitching of Nonlinear Vibration Absorbers. 199 3.3.1 The Mechanical Model . . . . . . . . . . . . . 199 3.3.2 A More General Class of Problems . . . . . . . 200

3.4 Mechanical Systems With Widely Spaced Frequencies . 202 3.4.1 A Two-Mode Model. . . . . . . . 203 3.4.2 The Geometry of Energy Transfer 204 3.4.3 An Example . . . . . . . . . . . . . 207

3.5 Irregular Particle Motion in the Atmosphere 211 3.5.1 The Model. . . . . . . . . . . . . . 211 3.5.2 Phase Space Geometry and Its Physical Meaning 212

xiv Contents

3.6 Subhannonic Generation in an Optical Cavity 3.6.1 A Two-Mode Model .... . 3.6.2 The Ideal Cavity (10 = 0) ...... . 3.6.3 Chaotic Dynamics for 10 > 0 .... .

3.7 Intennittent Bursting in Turbulent Boundary Layers 3.7.1 Modal Equations With Weak 0(2) ---+ D4

Symmetry Breaking . . . 3.7.2 The Slow Manifold ... 3.7.3 Fast Heteroclinic Cycles

3.8 Further Problems ...... .

4 Resonances in Hamiltonian Systems 4.1 Resonant Equilibria . . . . . . . . . . .

4.1.1 Birkhoff Nonnal Fonn ..... 4.1.2 A Class of 1 : 2 : k Resonances . 4.1.3 Geometry of the Normal Fonn 4.1.4 Homoclinic Orbits in the Two-Degree-of-Freedom

Subsystem ..................... . 4.1.5 Homoclinic Jumping in the Nonnal Fonn .... . 4.1.6 Homoclinic Jumping and Chaos in the Full Problem.

4.2 The Classical Water Molecule . . . . . . . . . . 4.2.1 The Nonnal Fonn . . . . . . . . . . . . 4.2.2 Homoclinic Chaos and Energy Transfer

4.3 Dynamics Near Intersecting Resonances . . . . 4.3.1 Arnold Diffusion in Near-Integrable Systems 4.3.2 Cross-Resonance Diffusion ......... . 4.3.3 Nonnal Fonn for Weak-Strong Double Resonances 4.3.4 The Pendulum-Type Hamiltonian .. 4.3.5 Dynamics in the Full Nonnal Fonn .

4.4 An Example From Rigid Body Dynamics. 4.5 Resonances in A Priori Unstable Systems.

4.5.1 A Physical Example ....... . 4.5.2 Whiskered Tori ......... . 4.5.3 Resonances on Invariant Manifolds. 4.5.4 Cross-Resonance Diffusion, Homoclinic Bifurcations,

and Horseshoes . . . . . . . . . . . . . . . . . . . . .

5 Chaotic Jumping Near Resonances: Infinite-Dimensional Systems 5.1 The Main Examples ................. . 5.2 Assumptions and Definitions .... . . . . . . . . .

5.2.1 The Phase Space and the Evolution Equation. 5.2.2 Regularity and Geometric Assumptions 5.2.3 N -Chains of Homoclinic Orbits

5.3 Invariant Manifolds and Foliations 5.3.1 Partially Slow Manifold .....

215 215 217 218 220

223 224 225 228

231 231 232 233 235

241 244 245 248 250 251 255 255 257 258 262 269 271 277 278 280 282

284

286 287 288 288 289 295 298 298

5.4

5.5

5.6

5.7

5.8 5.9 5.10

5.11

5.12

5.13

5.14

Contents xv

5.3.2 Preliminary Normal Form ...... . 5.3.3 Smooth Foliations for Wi~c(ME,k) and

Wl~c(ME.k) ........ . 5.3.4 N-Pulse Homoclinic Orbits. Passage Lemmas . . . . . . . . . . . 5.4.1 Fenichel Normal Form ... 5.4.2 Entry Conditions and Passage Time 5.4.3 Local Estimates Tracking Lemmas . . . . 5.5.1 The Local Map 5.5.2 The Global Map. Energy Lemmas . . . . . 5.6.1 Energy as a Coordinate 5.6.2 Energy of Entry Points 5.6.3 Improved Local Estimates 5.6.4 Energy of Projected Entry Points Multipulse Homoclinic Orbits in Sobolev Spaces. 5.7.1 Definitions and Notation ........ . 5.7.2 Existence Theorem ........... . 5.7.3 Remarks on Applications ofthe Main Theorem 5.7.4 Chain-Independent EnergyFunctions ..... . 5.7.5 Formulation With Other Invariants ...... . Disintegration of Invariant Manifolds Through Jumping Generalized Silnikov Orbits . . . . . . . . The Purely Hamiltonian Case . . . . . . . . . . . . . 5.10.1 Universal Homoclinic Bifurcations ..... . Homoclinic Jumping in the Perturbed NLS Equation. 5.11.1 Homoclinic Tree in the Forced NLS

Equation (D == 0) . . . . . . . . . . . 5.11.2 N -Pulse Orbits in the Damped-Forced

NLS Equation . . . . . . . . . . . . . 5.11.3 Silnikov-Type Orbits in the Damped-Forced

NLS Equation . . . . . . . . . . . . . . . Partially Slow Manifolds of Higher Codimension 5.12.1 Setup .................. . 5.12.2 Existence Theorem for Multipulse Orbits. 5.12.3 Multipulse Silnikov Manifolds ..... . Homoclinic Jumping in the CNLS System . . . . 5.13.1 Homoclinic Jumping in the Forced CNLS Equations

(ak = fh = 0) .................. . 5.13.2 N -Pulse Jumping Orbits in the Damped-Forced

CNLS System ({3k = 0) ..... . 5.13.3 N -Pulse Silnikov Manifolds in the

Full CNLS System Bibliographical Notes . . . . . . . . . . .

304

306 309 310 311 314 316 318 319 321 321 322 323 326 328 329 329 330 336 336 336 337 338 341 342 342

342

346

348 354 354 358 362 363

364

365

366 369

xvi Contents

A Elements of Differential Geometry Al Manifolds................. A2 Tangent, Cotangent, and Normal Bundles . A3 Transversality....... A4 Maps on Manifolds . . . . A5 Regular and Critical Points A6 Lie Derivative . . . . . . . A.7 Lie Algebras, Lie Groups, and Their Actions A.8 Orbit Spaces . . . . . . . . . . A9 Infinite-Dimensional Manifolds AIO Differential Forms . . . . . . All Maps and Differential Forms A12 Exterior Derivative .... A.13 Closed and Exact Forms . . . AI4 Lie Derivative of Forms . . . A.15 Volume Forms and Orientation AI6 Symplectic Forms . . . . . . . A.17 Poisson Brackets . . . . . . . . AI8 Integration on Manifolds and Stokes's Theorem

B Some Facts From Analysis B.I Fourier Series . . . . B.2 Gronwall Inequality B.3 Banach and Hilbert Spaces BA Differentiation and the Mean Value Theorem B.5 Distributions and Generalized Derivatives B.6 Sobolev Spaces ........... . B.7 Coo Bump Functions ......... . B.8 Factorization of Functions With a Zero

References

Symbol Index

Index

371 371 372 374 375 377 377 378 379 380 381 383 384 385 385 385 386 387 388

390 390 392 393 394 396 397 398 399

401

421

423