Ordinary Differential Equations and Dynamical Systems Gerald Teschl Graduate Studies in Mathematics Volume 140 American Mathematical Society http://dx.doi.org/10.1090/gsm/140 Ordinary Differential Equations and Dynamical Systems Gerald Teschl Graduate Studies in Mathematics Volume 140 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 34-01, 37-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-140 Library of Congress Cataloging-in-Publication Data Teschl, Gerald, 1970– Ordinary differential equations and dynamical systems / Gerald Teschl. p. cm. — (Graduate studies in mathematics ; v. 140) Includes bibliographical references and index. ISBN 978-0-8218-8328-0 (alk. paper) 1. Differential equations—Textbooks. 2. Dynamics—Textbooks. I. Title. QA371.T43 2011 515.352—dc23 2012015024 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 2012 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 171615141312 To Susanne, Simon, and Jakob Contents Preface ix Part 1. Classical theory Chapter 1. Introduction 3 §1.1. Newton’s equations 3 §1.2. Classification of differential equations 6 §1.3. First-order autonomous equations 9 §1.4. Finding explicit solutions 13 §1.5. Qualitative analysis of first-order equations 20 §1.6. Qualitative analysis of first-order periodic equations 28 Chapter 2. Initial value problems 33 §2.1. Fixed point theorems 33 §2.2. The basic existence and uniqueness result 36 §2.3. Some extensions 39 §2.4. Dependence on the initial condition 42 §2.5. Regular perturbation theory 48 §2.6. Extensibility of solutions 50 §2.7. Euler’s method and the Peano theorem 54 Chapter 3. Linear equations 59 §3.1. The matrix exponential 59 §3.2. Linear autonomous first-order systems 66 §3.3. Linear autonomous equations of order n 74 v vi Contents §3.4. General linear first-order systems 80 §3.5. Linear equations of order n 87 §3.6. Periodic linear systems 91 §3.7. Perturbed linear first-order systems 97 §3.8. Appendix: Jordan canonical form 103 Chapter 4. Differential equations in the complex domain 111 §4.1. The basic existence and uniqueness result 111 §4.2. The Frobenius method for second-order equations 116 §4.3. Linear systems with singularities 130 §4.4. The Frobenius method 134 Chapter 5. Boundary value problems 141 §5.1. Introduction 141 §5.2. Compact symmetric operators 146 §5.3. Sturm–Liouville equations 153 §5.4. Regular Sturm–Liouville problems 155 §5.5. Oscillation theory 166 §5.6. Periodic Sturm–Liouville equations 175 Part 2. Dynamical systems Chapter 6. Dynamical systems 187 §6.1. Dynamical systems 187 §6.2. The flow of an autonomous equation 188 §6.3. Orbits and invariant sets 192 §6.4. The Poincar´e map 197 §6.5. Stability of fixed points 198 §6.6. Stability via Liapunov’s method 201 §6.7. Newton’s equation in one dimension 203 Chapter 7. Planar dynamical systems 209 §7.1. Examples from ecology 209 §7.2. Examples from electrical engineering 215 §7.3. The Poincar´e–Bendixson theorem 220 Chapter 8. Higher dimensional dynamical systems 229 §8.1. Attracting sets 229 §8.2. The Lorenz equation 234 Contents vii §8.3. Hamiltonian mechanics 238 §8.4. Completely integrable Hamiltonian systems 243 §8.5. The Kepler problem 247 §8.6. The KAM theorem 250 Chapter 9. Local behavior near fixed points 255 §9.1. Stability of linear systems 255 §9.2. Stable and unstable manifolds 257 §9.3. The Hartman–Grobman theorem 264 §9.4. Appendix: Integral equations 270 Part 3. Chaos Chapter 10. Discrete dynamical systems 281 §10.1. The logistic equation 281 §10.2. Fixed and periodic points 284 §10.3. Linear difference equations 287 §10.4. Local behavior near fixed points 288 Chapter 11. Discrete dynamical systems in one dimension 293 §11.1. Period doubling 293 §11.2. Sarkovskii’s theorem 296 §11.3. On the definition of chaos 297 §11.4. Cantor sets and the tent map 300 §11.5. Symbolic dynamics 303 §11.6. Strange attractors/repellers and fractal sets 309 §11.7. Homoclinic orbits as source for chaos 313 Chapter 12. Periodic solutions 317 §12.1. Stability of periodic solutions 317 §12.2. The Poincar´e map 319 §12.3. Stable and unstable manifolds 321 §12.4. Melnikov’s method for autonomous perturbations 324 §12.5. Melnikov’s method for nonautonomous perturbations 329 Chapter 13. Chaos in higher dimensional systems 333 §13.1. The Smale horseshoe 333 §13.2. The Smale–Birkhoff homoclinic theorem 335 §13.3. Melnikov’s method for homoclinic orbits 336 viii Contents Bibliographical notes 341 Bibliography 345 Glossary of notation 349 Index 351 Preface About When you publish a textbook on such a classical subject the first ques- tion you will be faced with is: Why another book on this subject? Ev- erything started when I was supposed to give the basic course on Ordinary Differential Equations in Summer 2000. (At that time the course met 5 hours per week.) While there were many good books on the subject available, none of them quite fit my needs. I wanted a concise but rigorous introduction with full proofs that also covered classical topics such as Sturm–Liouville bound- ary value problems, differential equations in the complex domain, as well as modern aspects of the qualitative theory of differential equations. The course was continued with a second part on Dynamical Systems and Chaos in Winter 2000/01, and the notes were extended accordingly. Since then the manuscript has been rewritten and improved several times according to the feedback I got from students over the years when I redid the course. Moreover, since I had the notes on my homepage from the very beginning, this triggered a significant amount of feedback as well, from students who reported typos, incorrectly phrased exercises, etc., to colleagues who re- ported errors in proofs and made suggestions for improvements, to editors who approached me about publishing the notes. All this interest eventually resulted in a Chinese translation of an earlier version of the book. Moreover, if you google for the manuscript, you can see that it is used at several places worldwide, linked as a reference at various sites, including Wikipedia. Fi- nally, Google Scholar will tell you that it is even cited in several publications. Hence I decided that it was time to turn it into a real book. ix x Preface Content This book’s main aim is to give a self-contained introduction to the field of ordinary differential equations with emphasis on the dynamical systems point of view while still keeping an eye on classical tools as pointed out before. The first part is what I typically cover in the introductory course for bachelor’s level students. Of course it is typically not possible to cover ev- erything and one has to skip some of the more advanced sections. Moreover, it might also be necessary to add some material from the first chapter of the second part to meet curricular requirements. The second part is a natural continuation beginning with planar exam- ples (culminating in the generalized Poincar´e–Bendixson theorem), continu- ing with the fact that things get much more complicated in three and more dimensions, and ending with the stable manifold and the Hartman–Grobman theorem. The third and last part gives a brief introduction to chaos, focusing on two selected topics: Interval maps with the logistic map as the prime example plus the identification of homoclinic orbits as a source for chaos and the Melnikov method for perturbations of periodic orbits and for finding homoclinic orbits. Prerequisites This book requires only some basic knowledge of calculus, complex func- tions, and linear algebra. In addition, I have tried to show how a computer system, Mathematica1, can help with the investigation of differential equa- tions. However, the course is not tied to Mathematica and any similar pro- gram can be used as well. Updates The AMS is hosting a Web page for this book at http://www.ams.org/bookpages/gsm-140/ where updates, corrections, and other material may be found, including a link to material on my website: http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ 1Mathematica is a registered trademark of Wolfram Research, Inc. Preface xi There you can also find an accompanying Mathematica notebook with the code from the text plus some additional material. Acknowledgments I wish to thank my students, Ada Akerman, Kerstin Ammann, J¨org Arnberger, Alexander Beigl, Paolo Capka, Jonathan Eckhardt, Michael Fis- cher, Anna Geyer, Ahmed Ghneim, Hannes Grimm-Strele, Tony Johansson, Klaus Kr¨oncke, Alice Lakits, Simone Lederer, Oliver Leingang, Johanna Michor, Thomas Moser, Markus M¨uller, Andreas N´emeth, Andreas Pich- ler, Tobias Preinerstorfer, Jin Qian, Dominik Rasipanov, Martin Ringbauer, Reinaldo Garcia Rosario, Simon R¨oßler, Robert Stadler, Shelby Stanhope, Raphael Stuhlmeier, Gudrun Szewieczek, Gerhard Tulzer, Paul Wedrich, Florian Wisser, and colleagues, Edward Dunne, Klemens Fellner, Giuseppe Ferrero, Ilse Fischer, Delbert Franz, Heinz Hanßmann, Daniel Lenz, Jim Sochacki, and Eric Wahl´en, who have pointed out several typos and made useful suggestions for improvements.