常微分方程与动力系统

Preface Part 1. Classical theory Chapter 1. Introduction §1.1. Newton's equations §1.2. Classification of differential equations §1.3. First-order autonomous equations §1.4. Finding explicit solutions §1.5. Qualitative analysis of first-order equations §1.6. Qualitative analysis of first-order periodic equations Chapter 2. Initial value problems §2.1. Fixed point theorems §2.2. The basic existence and uniqueness result §2.3. Some extensions §2.4. Dependence on the initial condition §2.5. Regular perturbation theory §2.6. Extensibility of solutions §2.7. Euler's method and the Peano theorem Chapter 3. Linear equations §3.1. The matrix exponential §3.2. Linear autonomous first-order systems §3.3. Linear autonomous equations of order n §3.4. General linear first-order systems §3.5. Linear equations of order n §3.6. Periodic linear systems §3.7. Perturbed linear first-order systems §3.8. Appendix:Jordan canonical form Chapter 4. Differential equations in the complex domain §4.1. The basic existence and uniqueness result §4.2. The Frobenius method for second-order equations §4.3. Linear systems with singularities §4.4. The Frobenius method Chapter 5. Boundary value problems §5.1. Introduction §5.2. Compact symmetric operators §5.3. Sturm-Liouville equations §5.4. Regular Sturm-Liouville problems §5.5. Oscillation theory §5.6. Periodic Sturm-Liouville equations Part 2. Dynamical systems Chapter 6. Dynamical systems §6.1. Dynamical systems §6.2. The flow of an autonomous equation §6.3. Orbits and invariant sets §6.4. The Poincaré map §6.5. Stability of fixed points §6.6. Stability via Liapunov's method §6.7. Newton's equation in one dimension Chapter 7. Planar dynamical systems §7.1. Examples from ecology §7.2. Examples from electrical engineering §7.3. The Poincaré-Bendixson theorem Chapter 8. Higher dimensional dynamical systems §8.1. Attracting sets §8.2. The Lorenz equation §8.3. Hamiltonian mechanics §8.4. Completely integrable Hamiltonian systems §8.5. The Kepler problem §8.6. The KAM theorem Chapter 9. Local behavior near fixed points §9.1. Stability of linear systems §9.2. Stable and unstable manifolds §9.3. The Hartman-Grobman theorem §9.4. Appendix: Integral equations Part 3. Chaos Chapter 10. Discrete dynamical systems §10.1. The logistic equation §10.2. Fixed and periodic points §10.3. Linear difference equations §10.4. Local behavior near fixed points Chapter 11. Discrete dynamical systems in one dimension §11.1. Period doubling §11.2. Sarkovskii's theorem §11.3. On the definition of chaos §11.4. Cantor sets and the tent map §11.5. Symbolic dynamics §11.6. Strange attractors/repellers and fractal sets §11.7. Homoclinic orbits as source for chaos Chapter 12. Periodic solutions §12.1. Stability of periodic solutions §12.2. The Poincaré map §12.3. Stable and unstable manifolds §12.4. Melnikov's method for autonomous perturbations §12.5. Melnikov's method for nonautonomous perturbations Chapter 13. Chaos in higher dimensional systems §13.1. The Smale horseshoe §13.2. The Smale-Birkhoff homoclinic theorem §13.3. Melnikov's method for homoclinic orbits Bibliographical notes Bibliography Glossary of notation Index