Ordinary differential equations and dynamical systems

370 Pages·2012·2.696 MB·English

by  Teschl, Gerald

Ordinary Differential
Equations and
Dynamical Systems
Gerald Teschl
Graduate Studies
in Mathematics
Volume 140
American Mathematical Society
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 informationand updates on this book, visit
www.ams.org/bookpages/gsm-140
Library of Congress Cataloging-in-Publication Data
Teschl,Gerald,1970–
Ordinarydifferentialequationsanddynamicalsystems/GeraldTeschl.
p.cm. —(Graduatestudiesinmathematics;v.140)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-8328-0(alk.paper)
1.Differentialequations—Textbooks. 2.Dynamics—Textbooks. I.Title.
QA371.T43 2011
515(cid:2).352—dc23
2012015024
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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
DifferentialEquationsinSummer2000. (Atthattimethecoursemet5hours
perweek.) Whilethereweremanygoodbooksonthesubjectavailable,none
ofthemquitefitmyneeds. Iwantedaconcisebutrigorousintroductionwith
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
resultedinaChinesetranslationofanearlierversionofthebook. 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,GoogleScholarwilltellyouthatitisevencitedinseveralpublications.
Hence I decided that it was time to turn it into a real book.
ix