Chapter 1 The High Dimensional Melnikov Method with Nonresonance 1
1.1 Introduction 1
1.2 The geometric structure of the unperturbed System I 4
1.3 The geometric structure of the perturbed phase space 8
Chapter 2 On the Persistence of Lower Dimensional Invariant Tori for Hamiltonian Systems 15
2.1 Introduction 15
2.2 Some elementary reviews 18
2.3 Main results and proof 22
2.4 Time-dependent quasi-periodic perturbation 32
Chapter 3 Some Dynamics in ABC Flow 35
3.1 Introduction 35
3.2 Structure of the ABC flow with C=0 36
3.3 Invariant tori in the ABC flow with C≠0 38
3.4 Chaotic streamlines in the ABC flow with C≠0 43
Chapter 4 Geometric Singular Perturbation Theory with Resonance 46
4.1 Introduction 46
4.2 The set up 47
4.3 Main results 51
Chapter 5 Spectral Approximation of Attractors for the Damped and Driven Periodically Sine-Gordon Equation 57
5.1 Introduction 57
5.2 General approximation results 59
5.3 The existence of a global attractor 63
5.4 Attractors under the spectral approximation and their convergence 73
Chapter 6 Mechanism Causing Chaotic Jumping Behavior in the Damped Driven Sine-Gordon Equation 81
6.1 Introduction 81
6.2 The ODE on GAIM of the sine-Gordon equation 83
6.3 Unperturbed structure 87
6.4 Perturbed structure and dynamics near the resonance 91
6.5 Homocline orbits and pulse orbits 93
Chapter 7 Exact Solutions of the Nonlinear Evolution Equations 102
7.1 Introduction 102
7.2 Frame of the method 104
7.3 Two examples 107
7.4 A family of interesting exact solutions of the sine-Gordon equation 113
References 122
Acknowledgement 126