ODE and dynamical systems (lecture notes) PDF
Preview ODE and dynamical systems (lecture notes)
Ordinary differential equations and Dynamical Systems Gerald Teschl Gerald Teschl Fakult¨at fu¨r Mathematik Nordbergstraße 15 Universit¨at Wien 1090 Wien, Austria E-mail: [email protected] URL: http://www.mat.univie.ac.at/˜gerald/ 1991 Mathematics subject classification. 34-01 Abstract. Thismanuscriptprovidesanintroductiontoordinarydifferential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous linear flow. ThenweestablishtheFrobeniusmethodforlinearequationsinthecom- plexdomainandinvestigatesSturm–Liouvilletypeboundaryvalueproblems including oscillation theory. Next we introduce the concept of a dynamical system and discuss sta- bility including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems. We prove the Poincar´e–Bendixson theorem and investigate several ex- amples of planar systems from classical mechanics, ecology, and electrical engineering. Moreover,attractors,Hamiltoniansystems,theKAMtheorem, and periodic solutions are discussed as well. Finally, there is an introduction to chaos. Beginning with the basics for iterated interval maps and ending with the Smale–Birkhoff theorem and the Melnikov method for homoclinic orbits. Keywords and phrases. Ordinary differential equations, dynamical systems, Sturm-Liouville equations. Typeset by AMS-LATEX and Makeindex. Version: May 6, 2007 Copyright (cid:13)c 2000-2007 by Gerald Teschl Contents Preface vii 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 8 §1.4. Finding explicit solutions 12 §1.5. Qualitative analysis of first order equations 18 Chapter 2. Initial value problems 25 §2.1. Fixed point theorems 25 §2.2. The basic existence and uniqueness result 28 §2.3. Dependence on the initial condition 31 §2.4. Extensibility of solutions 34 §2.5. Euler’s method and the Peano theorem 37 §2.6. Appendix: Volterra integral equations 40 Chapter 3. Linear equations 45 §3.1. The matrix exponential 45 §3.2. Linear autonomous first order systems 50 §3.3. Linear autonomous equations of order n 53 §3.4. General linear first order systems 58 §3.5. Periodic linear systems 63 iii iv Contents §3.6. Appendix: Jordan canonical form 67 Chapter 4. Differential equations in the complex domain 73 §4.1. The basic existence and uniqueness result 73 §4.2. The Frobenius method for second order equations 76 §4.3. Linear systems with singularities 86 §4.4. The Frobenius method 89 Chapter 5. Boundary value problems 97 §5.1. Introduction 97 §5.2. Symmetric compact operators 100 §5.3. Regular Sturm-Liouville problems 105 §5.4. Oscillation theory 110 Part 2. Dynamical systems Chapter 6. Dynamical systems 119 §6.1. Dynamical systems 119 §6.2. The flow of an autonomous equation 120 §6.3. Orbits and invariant sets 123 §6.4. The Poincar´e map 127 §6.5. Stability of fixed points 128 §6.6. Stability via Liapunov’s method 130 §6.7. Newton’s equation in one dimension 132 Chapter 7. Local behavior near fixed points 137 §7.1. Stability of linear systems 137 §7.2. Stable and unstable manifolds 139 §7.3. The Hartman-Grobman theorem 144 §7.4. Appendix: Hammerstein integral equations 148 Chapter 8. Planar dynamical systems 151 §8.1. The Poincar´e–Bendixson theorem 151 §8.2. Examples from ecology 155 §8.3. Examples from electrical engineering 159 Chapter 9. Higher dimensional dynamical systems 165 §9.1. Attracting sets 165 §9.2. The Lorenz equation 168 §9.3. Hamiltonian mechanics 172 Contents v §9.4. Completely integrable Hamiltonian systems 176 §9.5. The Kepler problem 181 §9.6. The KAM theorem 183 Part 3. Chaos Chapter 10. Discrete dynamical systems 189 §10.1. The logistic equation 189 §10.2. Fixed and periodic points 192 §10.3. Linear difference equations 194 §10.4. Local behavior near fixed points 196 Chapter 11. Periodic solutions 199 §11.1. Stability of periodic solutions 199 §11.2. The Poincar´e map 200 §11.3. Stable and unstable manifolds 202 §11.4. Melnikov’s method for autonomous perturbations 205 §11.5. Melnikov’s method for nonautonomous perturbations 210 Chapter 12. Discrete dynamical systems in one dimension 213 §12.1. Period doubling 213 §12.2. Sarkovskii’s theorem 216 §12.3. On the definition of chaos 217 §12.4. Cantor sets and the tent map 220 §12.5. Symbolic dynamics 223 §12.6. Strange attractors/repellors and fractal sets 228 §12.7. Homoclinic orbits as source for chaos 232 Chapter 13. Chaos in higher dimensional systems 237 §13.1. The Smale horseshoe 237 §13.2. The Smale-Birkhoff homoclinic theorem 239 §13.3. Melnikov’s method for homoclinic orbits 240 Bibliography 245 Glossary of notations 247 Index 248 Index 249 Preface The present manuscript constitutes the lecture notes for my courses Ordi- nary Differential Equations and Dynamical Systems and Chaos held at the University of Vienna in Summer 2000 (5hrs.) and Winter 2000/01 (3hrs.), respectively. It is supposed to give a self contained introduction to the field of ordi- nary differential equations with emphasize on the view point of dynamical systems. Itonlyrequiressomebasicknowledgefromcalculus, complexfunc- tions,andlinearalgebrawhichshouldbecoveredintheusualcourses. Itried to show how a computer system, Mathematica, can help with the investiga- tion of differential equations. However, any other program can be used as well. The manuscript is available from http://www.mat.univie.ac.at/˜gerald/ftp/book-ode/ Acknowledgments Iwishtothankmystudents,K.Ammann,P.Capka,A.Geyer,J.Michor, F. Wisser, and colleagues, K. Fellner and D. Lenz, who have pointed out several typos and made useful suggestions for improvements. Gerald Teschl Vienna, Austria January, 2007 vii Part 1 Classical theory
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