Series Preface
Preface to the Second Edition
Introduction
1 Equilibrium Solutions，Stability,and Linearized Stability
1.1 Equilibria of Vector Fields
1.2 Stability of Trajectories
1.2a Linearization
1.3 Maps。
1.3a Definitions of Stability for Maps
1.3b Stability of Fixed Points of Linear Maps
1.3c Stability of Fixed Points of Maps via the Linear
Approximation
1.4 Some Terminology Associated with Fixed Points
1.5 Application to the Unforced Duffing Oscillator
1.6 Exercises
2 Liapunov Functions
2.1 Exercises
3 Invariant Manifolds：Linear and Nonlinear Systems
3.1 Stable，Unstable，and Center Subspaces of Linear，Autonomous
Vector Fields
3.1a Invariance of the Stable，Unstable，and Center Subspaces
3.1b Some Examples.
3.2 Stable，Unstable，and Center Manifolds for Fixed Points of
Nonlinear，Autonomous Vector Fields
3.2a Invariance of the Graph of a Function：Tangency of the Vector
Field to the Graph
3.3 Maps
3.4 Some Examples
3.5 Existence of Invariant Manifolds：The Main Methods of Proof,and
HOW They Wbrk 3：5a Application of These Two Methods to a Concrete
Example：Existence of the Unstable Manifold
3.6 Time-Dependent Hyperbolic Trajectories and their Stable and
Unstable ManifoIds
3.6a Hyperbolic Trajectories
3.6b Stable and Unstable Manifolds of Hyperbolic Trajectories
3.7 Invariant Manifolds in a Broader Context
3.8 Exercises
4 Periodic Orbits
4.1 Nonexistence of Periodic Orbits for Two-Dimensional，Autonomous
Vector Fields
4.2 Further Remarks on Periodic Orbits
4.3 Exercises
5 Vector Fields Possessing an Integral
5.1 Vector Fields on Two-Manifolds Having an Integral
5.2 Two Degree-of-Freedom Hamiltonian Systems and Geometry
5.2a Dynamics on the Energy Surface.
5.2b Dynamics on an Individual Torus
5.3 Exercises
6 Index Theory
6.1 Exercises
7 Some General Properties of Vector Fields：
Existence，Uniqueness，Differentiability,and Flows
7.1 Existence，Uniqueness，Differentiability with Respect to Initial
Conditions
7.2 Continuation of Solutions
7.3 Differentiability with Respect to Parameters
7.4 Autonomous Vector Fields
7.5 Nonautonomous Vector Fields
7.5a The Skew—Product Flow Approach
7.5b The Cocycle Approach
7.5c Dynamics Generated by a Bi—Infinite Sequence of Maps
7.6 Liouville''s Theorem
7.6a Volume Preserving Vector Fields and the Poincar6 Recurrence
Theorem
7.7 Exercises
8 Asymptotic Behavior
8.1The Asymptotic Behavior ofTrajectories.
8.2 Attracting Sets，Attractors.and Basins of Attraction
8.3 The LaSalle Invariance Principle
8.4 Attraction in Nonautonomous Systems
8.5 Exercises
9 The Poinear6-Bendixson Theorem
9.1 Exercises
10 Poinear6 Maps
10.1 Cuse 1：Poincar6 Map Near a Periodic Orbit
10.2 Case 2：The Poincar6 Map of a Time-Periodic Ordinary
Differential Equation
10.2a Periodically Forced Linear Oscillators
10.3 Case 3：The Poincar6 MaD Near a Homoclinic Orbit
10.4 Case 4：Poincar6 Map Associated with a Two Degree-of-Freedom
Hamiltonian System
10.4a The Study of Coupled Oscillators via Circle Maps
10.5 Exercises
11 Conjugacies of Maps，and Varying the Cross.Section
11.1 Case 1：Poincar6 Map Near a Periodic Orbit：Variation of the
Cross—Section
11.2 Case 2：The Poincard Map of a Time-Periodic Ordinary
Differential Equation：Variation of the Cross—Section
12 Structural Stability，Genericity，and Transversality
12.1 Definitions of Structural Stability and Genericity。
12.2 Transversality
12.3 Exercises。。
13 Lagrange''s Equations
13.1 Generalized Coordinates
13.2 Derivation of Lagrange''s Equations
13.2a The Kinetic Energy
13.3 The Energy Integral
13.4 Momentum Integrals
13.5 Hamilton''s Equations
13.6 Cyclic Coordinates.Routh''s Equations.and Reduction of the
Number of Equations
13.7 Variational Methods
13.7a The Principle of Least Action
13.7b The Action Principle in Phase Space。
13.7c Transformations that Preserve the Form of Hamilton''s
Equations
13.7d Applications of Variational Methods
13.8 The Hamilton.Jacobi Equation
13.8a Applications of the Hamilton—Jacobi Equation
13.9 Exercises，
14 Hamiltonian vector Fields
14.1 Symplectic Forms
14.1a The Relationship Between Hamilton''s Equations and the
Symplectic Form
14.2 Poisson Brackets
14.2a Hamilton’s Equations in Poisson Bracket Form
14.3 Symplectic or Canonical Transformations.
14.3a Eigenvalues of Symplectic Matrices
14.3b Infinitesimally Symplectic Transformations
14.3c The Eigenvalues of Infinitesimally Symplectic Matrices
14.3d The Flow Generated by Hamiltonian Vector Fields is a
One-Parameter FamiIy of Symplectic Transformations
14.4 Transformation of Hamilton''s Equations Under Symplectic
Transformations
14.4a Hamilton''s Equations in Complex Coordinates
14.5 Completely Integrable Hamiltonian Systems
14.6 Dynamics of Completely Integrable Hamiltonian Systems in
Action—Angle Coordinates
14.6a Resonance and Nonresonance
14.6b Diophantine Frequencies
14.6c Geometry of the Resonances
14.7 Perturbations of Completely Integrable Hamiltonian Systems in
Action-Angle Coordinates
14.8 Stability of Elliptic Equilibria
14.9 Discrete-Time Hamiltonian Dynamical Systems：Iteration of
Symplectic Maps
14.9a The KAM Theorem and Nekhoroshev''s Theorem for Symplectic
Maps.
14.10 Genetic Properties of Hamiltonian Dynamical Systems
14.11 Exercises
15 Gradient vector Fields
15.1 Exercises
16 Reversible Dynamical Systems
16.1 The Definition of Reversible Dynamical Systems
16.2 Examples of Reversible Dynamical Systems
16.3 Linearization of Reversible Dynamical Systems
16.3a Continuous Time
16.3b Discrete Time
……
17 Asymptotically Autonomous Vector Fields
18 Center Manifolds
19 Normal Forms
20 Bifurcation of Fixed Points of Vector Fields
21 Bifurcations of Fixed Points of Maps
23 The Smale Horseshoe
24 Symbolic Dynamics
25 The Conley-Moser Conditions，or“How to Prove That a Dynamical
System is Chaotic”
26 Dynamics Near Homoclinic Points ofTwo-Dimensional Maps
27 Orbits Homoclinic to Hyperbolic Fixed Points in
Three-Dimensional Autonomous V.ector Fields
28 Melnikov''s Method for Homoclinic Orbits in
Two-Dimensional，Time-Periodic Vector Fields
29 Liapunov Exponents
30 Chaos and Strange Attractors
31 Hyperbolic Invariant Sets：A Chaotic Saddle
32 Long Period Sinks in Dissipative Systems and Elliptic Islands in
Conservative Systems
33 Global Bifurcations Arising from Local Codimension--Two
Bifurcations
34 Glossary of Frequently Used Terms
Bibliography
Index
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