Mathematical Surveys and Monographs Volume 176 Nonautonomous Dynamical Systems Peter E. Kloeden Martin Rasmussen American Mathematical Society http://dx.doi.org/10.1090/surv/176 Nonautonomous Dynamical Systems Mathematical Surveys and Monographs Volume 176 Nonautonomous Dynamical Systems Peter E. Kloeden Martin Rasmussen American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Jordan S. Ellenberg Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 37B55, 37C60, 37H05, 37B25, 37C75, 37D10, 37G35. For additional information and updates on this book, visit www.ams.org/bookpages/surv-176 Library of Congress Cataloging-in-Publication Data Kloeden, Peter E. Nonautonomous dynamical systems / Peter E. Kloeden, Martin Rasmussen. p. cm. — (Mathematical surveys and monographs ; v. 176) Includes bibliographical references and index. ISBN 978-0-8218-6871-3 (alk. paper) 1. Dynamics. 2. Ergodic theory. 3. Stability. I. Rasmussen, Martin, 1975– II. Title. QA845.K56 2011 515.392—dc23 2011020550 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 Contents Preface vii Chapter 1. Autonomous dynamical systems 1 1. Introduction 1 2. Local asymptotic behavior 4 3. Global asymptotic behavior 12 4. Dependence on parameters 17 Chapter 2. Nonautonomous dynamical systems 23 1. Processes formulation 23 2. Skew product flow formulation 26 3. Entire solutions and invariant sets 31 Chapter 3. Attractors 37 1. Attractors of processes 38 2. Attractors of skew product flows 41 3. Existence of pullback attractors 44 4. Relationship between nonautonomous attractors 52 5. Upper semi-continuous dependence on parameters 55 6. Parametrically inflated pullback attractors 57 7. Pullback attractors with continuous fibers 60 8. Local attractors and repellers 62 Chapter 4. Morse decompositions 69 1. Attractor-repeller pairs 69 2. Morse decompositions 72 3. The one-dimensional case 75 Chapter5. Linearsystems 79 1. Exponential dichotomy 79 2. Dichotomy spectrum 82 3. Lyapunov spectrum 87 4. Morse decompositions 89 Chapter 6. Invariant manifolds 105 1. Global invariant manifolds 105 2. Local invariant manifolds 112 3. Hierarchies of invariant manifolds 114 4. Taylor approximation 116 5. Reduction principle 123 v vi CONTENTS Chapter 7. Lyapunov functions 129 1. Lyapunov functions for solutions 129 2. Lyapunov functions for autonomous attractors 132 3. Lyapunov functions for pullback attractors 135 4. Lyapunov functions for Morse decompositions 143 Chapter 8. Bifurcations 147 1. Nonautonomous Bernoulli equations 147 2. One-dimensional bifurcation patterns 149 3. Higher-dimensional Bernoulli-like equations 157 4. Further developments 163 Chapter 9. Set-valued nonautonomous dynamical systems 169 1. Set-valued processes 170 2. Set-valued skew product flows 173 3. Backward extension of autonomous semi-dynamical systems 175 4. Proof of existence of nonautonomous invariant sets 178 Chapter 10. Nonautonomous semi-dynamical systems 185 1. Attractors of skew product semi-flows 185 2. The twisted horseshoe mapping 189 Chapter 11. Approximation and perturbation of attractors 191 1. Nonautonomous perturbations of an autonomous system 191 2. Numerical approximation of uniform attractors 193 3. Perturbation of the driving system 197 Chapter 12. Infinite-dimensional systems 205 1. Squeezing and flattening properties: the autonomous case 205 2. Pullback asymptotic compactness 207 Chapter 13. Switching and control systems 213 1. Switching systems 213 2. Affine control systems 222 Chapter 14. Random dynamical systems 227 1. Random attractors 228 2. The Ornstein–Uhlenbeck process 229 3. Random attractors for stochastic differential equations 231 Chapter 15. Synchronization 235 1. Deterministic nonautonomous systems 235 2. Synchronization of systems with additive noise 242 3. Synchronization of systems with linear noise 247 Appendix 251 Bibliography 253 Index 263 Preface Dynamical systems theory has been a very active area of research in mathematics and cognate fields for many years, but most of the results that have been obtained focus almost exclusively on autonomous dynamical systems. There have, of course, been investigations of nonautonomous differential equations, that is with time- dependent vector fields, during this time, but it is only in the recent decade that a theory of nonautonomous dynamical systems has emerged synergizing parallel developments on time-dependent differential equations, control systems and ran- dom dynamical systems. There are now abstract formulations of nonautonomous dynamical systems as two-parameter semi-groups or processes and as skew product flows as well as new concepts of nonautonomous attractors, in particular, pullback attractors. This development is presented in this book for graduate students and others with a general background in dynamical systems and differential equations. The choice of topics and applications covered, especially in the later chapters, reflects the interests of the authors, but nevertheless provides a broad overview of important developments on the subject. There are fifteen chapters and an appendix. The first chapter briefly reviews the theory of autonomous dynamical systems from the perspective of what is needed later rather than attempting to be comprehensive in itself. The process and skew product flow formalism of nonautonomous dynamical systems are introduced in the second chapter and the various concepts of nonautonomous attractors are pre- sented and compared in the third chapter. These two chapters are essential reading for everything that is to follow. The remaining chapters can be read more or less independently of each other, except the fourth, fifth and sixth chapters on Morse de- compositions, linear systems and invariant manifolds, respectively, which are best read as a sequential block. Lyapunov functions are considered in chapter seven and bifurcations in nonautonomous systems in chapter eight. Generalizations to set-valued nonautonomous dynamical systems and nonautonomous semi-dynamical systems are treated in chapters nine and ten, while the effects of perturbations and discretization are discussed in chapter eleven. Up to here the state space is either Rd or a general complete metric space, but in chapter twelve issues of explicit relevance to infinite-dimensional state spaces are considered. Chapter thirteen applies previ- ous results to switching and affine control systems interpreted as nonautonomous dynamical systems, while chapter fourteen introduces readers to some of the differ- ences arising in random dynamical systems due to their measure-theoretic rather than topological characteristics. The previous deterministic and random results are then applied to the synchronization of dissipative systems in chapter fifteen. vii viii PREFACE Finally, various background definitions and results needed within the text are given in the appendix. Readers who are interested in the dynamical behavior of nonautonomous partial differential equations and evolution equations are advised to refer to the mono- graphs of Carvalho, Langa & Robinson [35]andChepyzhov & Vishik [43] in conjunction with this book. Acknowledgements. We are indebted to numerous colleagues for their help- ful discussions during the preparation of this book. In particular, we thank Tom´as Caraballo, Alexandre Carvalho, David Cheban, Hans Crauel, Jinquiao Duan, Mes- soud Effendiev, Barnabas Garay, Peter Giesl, Arnulf Jentzen, Victor Kozyakin, Jeroen Lamb, Jos´e Langa, Li Desheng, Thomas Lorenz, Pedro Mar´ın-Rubio, Chris- tian P¨otzsche, Jos´e Real, Janosch Rieger, James Robinson, Bj¨orn Schmalfuß, Ste- fanie Sonner, Aneta Stefanovska, Meihua Yang for carefully reading parts of the book and for their suggestions for improvements, as well as Fritz Colonius and Christoph Kawan for advice on control systems as skew product flows. In addi- tion, we thank Alexandre Carvalho, Jos´e Langa and James Robinson for keeping us informed about developments with preparation of their book [35] entitled Attrac- tors of Infinite Dimensional Nonautonomous Dynamical Systems and their useful comments about our manuscript. We also thank Sofie van Geene for providing Fig- ure 2.1 and Philipp Storck for providing Figures 3.1 and 3.2 in the book. Finally, we would like to thank both Karin and Eva-Maria for their encouragement during thetimewewrotethisbook. Peter Kloeden thanks the Departamento de Ecuaciones Diferenciales y An´alisis