Chapter 1 Qualitative Theory for ODE Systems

1.1 Basic notions

1.2 Local existence

1.2.1 Normed spaces and fixed point theorem

1.2.2 Applications to ODE system and linear algebraic system

1.3 Critical point

1.4 Plane analysis for the Duffing equation

1.5 Homoclinic orbit and limit cycle

1.6 Stability and Lyapunov function

1.7 Bifurcation

1.8 Chaos:Lorenz equations and logistic map

Chapter 2 Reaction-Diffusion Systems

2.1 Introduction:BVP and IBVP,equilibrium

2.2 Dispersion relation,linear and nonlinear stability

2.3 Invariant domain

2.4 Perturbation method

2.5 Traveling waves

2.6 Burgers'equation and Cole-Hopf transform

2.7 Evolutionary Duffing equation

Chapter 3 Elliptic Equations

3.1 Sobolev spaces

3.2 Variational formulation of second-order elliptic equations

3.3 Neumann boundary value problem

Chapter 4 Hyperbolic Conservation Laws

4.1 Linear advection equation,characteristics method

4.2 Nonlinear hyperbolic equations

4.3 Discontinuities in inviscid Burgers'equation

4.4 Elementary waves in inviscid Burgers'equation

4.5 Wave interactions in inviscid Burgers'equation

4.6 Elementary waves in a polytropic gas

4.7 Riemann problem in a polytropic gas

4.8 Elementary waves in a polytropic ideal gas

4.9 Soliton and inverse scattering transform

Index