Ordinary Differential Equations and Dynamical Systems

(Parte 1 de 9)

Ordinary Differential Equations and Dynamical Systems

Gerald Teschl

Gerald Teschl Fakultat fur Mathematik Nordbergstraße 15 Universitat Wien 1090 Wien, Austria

E-mail: Gerald.Teschl@univie.ac.at URL: http://www.mat.univie.ac.at/~gerald/

2000 Mathematics subject classification. 34-01

Abstract. This book provides an introduction to ordinary differential 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.

Then we establish the Frobenius method for linear equations in the complex domain and investigate Sturm–Liouville type boundary value problems including oscillation theory.

Next we introduce the concept of a dynamical system and discuss stability including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems.

We prove the Poincare–Bendixson theorem and investigate several examples of planar systems from classical mechanics, ecology, and electrical engineering. Moreover, attractors, Hamiltonian systems, the KAM theorem, 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: April 14, 2010

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 13

§1.5. Qualitative analysis of first-order equations 19

§1.6. Qualitative analysis of first-order periodic equations 26

§2.2. The basic existence and uniqueness result 34

§2.4. Dependence on the initial condition 39

§2.6. Euler’s method and the Peano theorem 47

Chapter 3. Linear equations 51 §3.1. The matrix exponential 51

§3.2. Linear autonomous first-order systems 56

§3.3. Linear autonomous equations of order n 62 iv Contents

§3.5. Periodic linear systems 75 §3.6. Appendix: Jordan canonical form 81

Chapter 4. Differential equations in the complex domain 87 §4.1. The basic existence and uniqueness result 87

§4.2. The Frobenius method for second-order equations 90

§4.3. Linear systems with singularities 101

Chapter 5. Boundary value problems 1 §5.1. Introduction 1

§5.2. Compact symmetric operators 114

§5.3. Regular Sturm-Liouville problems 120

§5.5. Periodic operators 133

Part 2. Dynamical systems

Chapter 6. Dynamical systems 145 §6.1. Dynamical systems 145

§6.2. The flow of an autonomous equation 146

§6.3. Orbits and invariant sets 149

§6.5. Stability of fixed points 155

§6.6. Stability via Liapunov’s method 156

§6.7. Newton’s equation in one dimension 159

Chapter 7. Local behavior near fixed points 165 §7.1. Stability of linear systems 165

§7.2. Stable and unstable manifolds 167

§7.3. The Hartman-Grobman theorem 174

§7.4. Appendix: Integral equations 179

Chapter 8. Planar dynamical systems 187 §8.1. The Poincare–Bendixson theorem 187

§8.2. Examples from ecology 191

§8.3. Examples from electrical engineering 196

(Parte 1 de 9)