Chapter 1 The High Dimensional Melnikov Method with Nonresonance

1.1 Introduction

1.2 The geometric structure of the unperturbed System I

1.3 The geometric structure of the perturbed phase space

Chapter 2 On the Persistence of Lower Dimensional Invariant Tori for Hamiltonian Systems

2.1 Introduction

2.2 Some elementary reviews

2.3 Main results and proof

2.4 Time-dependent quasi-periodic perturbation

Chapter 3 Some Dynamics in ABC Flow

3.1 Introduction

3.2 Structure of the ABC flow with C=0

3.3 Invariant tori in the ABC flow with C≠0

3.4 Chaotic streamlines in the ABC flow with C≠0

Chapter 4 Geometric Singular Perturbation Theory with Resonance

4.1 Introduction

4.2 The set up

4.3 Main results

Chapter 5 Spectral Approximation of Attractors for the Damped and Driven Periodically Sine-Gordon Equation

5.1 Introduction

5.2 General approximation results

5.3 The existence of a global attractor

5.4 Attractors under the spectral approximation and their convergence

Chapter 6 Mechanism Causing Chaotic Jumping Behavior in the Damped Driven Sine-Gordon Equation

6.1 Introduction

6.2 The ODE on GAIM of the sine-Gordon equation

6.3 Unperturbed structure

6.4 Perturbed structure and dynamics near the resonance

6.5 Homocline orbits and pulse orbits

Chapter 7 Exact Solutions of the Nonlinear Evolution Equations

7.1 Introduction

7.2 Frame of the method

7.3 Two examples

7.4 A family of interesting exact solutions of the sine-Gordon equation

References

Acknowledgement