前辅文
Preface
Part 1. Basic Concepts and Linear Equations
Chapter 1. Ordinary Differential Equations
§1.1. Basic notions
§1.2. Existence and uniqueness of solutions
§1.3. Additional properties
§1.4. Existence of solutions for continuous fields
§1.5. Phase portraits
§1.6. Equations on manifolds
§1.7. Exercises
Chapter 2. Linear Equations and Conjugacies
§2.1. Nonautonomous linear equations
§2.2. Equations with constant coefficients
§2.3. Variation of parameters formula
§2.4. Equations with periodic coefficients
§2.5. Conjugacies between linear equations
§2.6. Exercises
Part 2. Stability and Hyperbolicity
Chapter 3. Stability and Lyapunov Functions
§3.1. Notions of stability
§3.2. Stability of linear equations
§3.3. Stability under nonlinear perturbations
§3.4. Lyapunov functions
§3.5. Exercises
Chapter 4. Hyperbolicity and Topological Conjugacies
§4.1. Hyperbolic critical points
§4.2. The Grobman–Hartman theorem
§4.3. Hölder conjugacies
§4.4. Structural stability
§4.5. Exercises
Chapter 5. Existence of Invariant Manifolds
§5.1. Basic notions
§5.2. The Hadamard–Perron theorem
§5.3. Existence of Lipschitz invariant manifolds
§5.4. Regularity of the invariant manifolds
§5.5. Exercises
Part 3. Equations in the Plane
Chapter 6. Index Theory
§6.1. Index for vector fields in the plane
§6.2. Applications of the notion of index
§6.3. Index of an isolated critical point
§6.4. Exercises
Chapter 7. Poincaré–Bendixson Theory
§7.1. Limit sets
§7.2. The Poincaré–Bendixson theorem
§7.3. Exercises
Part 4. Further Topics
Chapter 8. Bifurcations and Center Manifolds
§8.1. Introduction to bifurcation theory
§8.2. Center manifolds and applications
§8.3. Theory of normal forms
§8.4. Exercises
Chapter 9. Hamiltonian Systems
§9.1. Basic notions
§9.2. Linear Hamiltonian systems
§9.3. Stability of equilibria
§9.4. Integrability and action-angle coordinates
§9.5. The KAM theorem
§9.6. Exercises
Bibliography
Index
