Ordinary Differential Equations and Dynamical Systems

Preface 8 Part 1. Classical theory 10 Chapter 1. Introduction 12 1.1. Newton's equations 12 1.2. Classification of differential equations 15 1.3. First order autonomous equations 17 1.4. Finding explicit solutions 22 1.5. Qualitative analysis of first-order equations 28 Chapter 2. Initial value problems 36 2.1. Fixed point theorems 36 2.2. The basic existence and uniqueness result 38 2.3. Some extensions 41 2.4. Dependence on the initial condition 43 2.5. Extensibility of solutions 48 2.6. Euler's method and the Peano theorem 50 Chapter 3. Linear equations 56 3.1. The matrix exponential 56 3.2. Linear autonomous first-order systems 61 3.3. Linear autonomous equations of order n 64 3.4. General linear first-order systems 70 3.5. Periodic linear systems 75 3.6. Appendix: Jordan canonical form 80 Chapter 4. Differential equations in the complex domain 84 4.1. The basic existence and uniqueness result 84 4.2. The Frobenius method for second-order equations 87 4.3. Linear systems with singularities 97 4.4. The Frobenius method 100 Chapter 5. Boundary value problems 106 5.1. Introduction 106 5.2. Symmetric compact operators 109 5.3. Regular Sturm-Liouville problems 114 5.4. Oscillation theory 119 5.5. Periodic operators 124 Part 2. Dynamical systems 134 Chapter 6. Dynamical systems 136 6.1. Dynamical systems 136 6.2. The flow of an autonomous equation 137 6.3. Orbits and invariant sets 140 6.4. The Poincaré map 144 6.5. Stability of fixed points 145 6.6. Stability via Liapunov's method 147 6.7. Newton's equation in one dimension 149 Chapter 7. Local behavior near fixed points 154 7.1. Stability of linear systems 154 7.2. Stable and unstable manifolds 156 7.3. The Hartman-Grobman theorem 161 7.4. Appendix: Integral equations 165 Chapter 8. Planar dynamical systems 174 8.1. The Poincaré--Bendixson theorem 174 8.2. Examples from ecology 178 8.3. Examples from electrical engineering 182 Chapter 9. Higher dimensional dynamical systems 188 9.1. Attracting sets 188 9.2. The Lorenz equation 191 9.3. Hamiltonian mechanics 195 9.4. Completely integrable Hamiltonian systems 199 9.5. The Kepler problem 204 9.6. The KAM theorem 206 Part 3. Chaos 210 Chapter 10. Discrete dynamical systems 212 10.1. The logistic equation 212 10.2. Fixed and periodic points 215 10.3. Linear difference equations 217 10.4. Local behavior near fixed points 219 Chapter 11. Periodic solutions 222 11.1. Stability of periodic solutions 222 11.2. The Poincaré map 223 11.3. Stable and unstable manifolds 225 11.4. Melnikov's method for autonomous perturbations 228 11.5. Melnikov's method for nonautonomous perturbations 233 Chapter 12. Discrete dynamical systems in one dimension 236 12.1. Period doubling 236 12.2. Sarkovskii's theorem 239 12.3. On the definition of chaos 240 12.4. Cantor sets and the tent map 243 12.5. Symbolic dynamics 246 12.6. Strange attractors/repellors and fractal sets 251 12.7. Homoclinic orbits as source for chaos 255 Chapter 13. Chaos in higher dimensional systems 260 13.1. The Smale horseshoe 260 13.2. The Smale-Birkhoff homoclinic theorem 262 13.3. Melnikov's method for homoclinic orbits 263 Bibliography 266 Glossary of notations 268 Index 270 Index 270 Chaos