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Chapter 1 Qualitative Theory for ODE Systems 1

1.1 Basic notions 1

1.2 Local existence 3

1.2.1 Normed spaces and fixed point theorem 4

1.2.2 Applications to ODE system and linear algebraic system 11

1.3 Critical point 14

1.4 Plane analysis for the Duffing equation 18

1.5 Homoclinic orbit and limit cycle 24

1.6 Stability and Lyapunov function 29

1.7 Bifurcation 33

1.8 Chaos:Lorenz equations and logistic map 38

Chapter 2 Reaction-Diffusion Systems 50

2.1 Introduction:BVP and IBVP,equilibrium 50

2.2 Dispersion relation,linear and nonlinear stability 57

2.3 Invariant domain 60

2.4 Perturbation method 63

2.5 Traveling waves 69

2.6 Burgers'equation and Cole-Hopf transform 72

2.7 Evolutionary Duffing equation 74

Chapter 3 Elliptic Equations 86

3.1 Sobolev spaces 86

3.2 Variational formulation of second-order elliptic equations 88

3.3 Neumann boundary value problem 93

Chapter 4 Hyperbolic Conservation Laws 95

4.1 Linear advection equation,characteristics method 95

4.2 Nonlinear hyperbolic equations 97

4.3 Discontinuities in inviscid Burgers'equation 101

4.4 Elementary waves in inviscid Burgers'equation 103

4.5 Wave interactions in inviscid Burgers'equation 107

4.6 Elementary waves in a polytropic gas 114

4.7 Riemann problem in a polytropic gas 121

4.8 Elementary waves in a polytropic ideal gas 126

4.9 Soliton and inverse scattering transform 128

Index 144