混沌初步

CHAPTER 1 Computational Techniques of Linear Differential Equation

1.1 Basic concepts

1.2 First order linear differential equation

1.2.1 Separable equation

1.2.2 Linear equation

1.2.3 Exact equations and integrating factors

1.2.4 Direction fields

1.3 Second order differential equation

1.3.1 Homogeneous linear equation

1.3.2 Nonhomogeneous linear equation

1.4 First order differential equations

1.4.1 Basic theories of the first order DEs

1.4.2 Homogeneous linear DEs with constant coefficients

1.4.3 Nonhomogeneous linear DEs with constant coefficients

1.5 Three special methods

CHAPTER 1  Computational Techniques of Linear Differential Equation

1.1  Basic concepts

1.2  First order linear differential equation

1.2.1  Separable equation

1.2.2  Linear equation

1.2.3  Exact equations and integrating factors

1.2.4  Direction fields

1.3  Second order differential equation

1.3.1  Homogeneous linear equation

1.3.2  Nonhomogeneous linear equation

1.4  First order differential equations

1.4.1  Basic theories of the first order DEs

1.4.2  Homogeneous linear DEs with constant coefficients

1.4.3  Nonhomogeneous linear DEs with constant coefficients

1.5  Three special methods

1.5.1  Laplace transform method

1.5.2  Power series method

1.5.3  Fourier series method

1.6  Numerical solution of differential equations

CHAPTER 2  Qualitative Analysis of Planar Differential Equations

2.1  Flow and manifold

2.1.1  Flow

2.1.2  Maniflod

2.2  Planar linear systems

2.3  Linearization of nonlinear systems

2.3.1  Singularities analysis of nonlinear systems

2.3.2  Stability of singularities

2.4  Periodic solutions of nonlinear systems

2.4.1  Orbit and limit set

2.4.2  Periodic orbit and limit cycle

2.5  Conservative system and dissipative system

2.5.1  Hamiltonian system

2.5.2  Dissipative systems

CHAPTER 3  Calculation and Analysis of Chaotic Systems

3.1  Attractor, Lyapunov exponent

3.1.1  Attractor

3.1.2  Lyapunov exponent

3.2  Center manifolds

3.2.1  Eigenspaces and manifolds

3.2.2  Center manifolds

3.3  Hopf bifurcation

3.3.1  Andmnov-Hopf bifurcation

3.3.2  Hopf bifurcation of Lorenz-like system

3.4  Dimension reduction analysis

3.4.1  Invariant algebraic surface

3.4.2  Invariant algebraic surface of T system

3.5  Infinity analysis

3.5.1  Poincare compactification on R2

3.5.2  Poincare compactification on R3

3.6  Melnikov method

CHAPTER 4  Control and Synchronization of Chaotic Systems

4.1  Feedback control

4.1.1  Feedback control of T system

4.1.2  Differential feedback control of Jerk system

4.2  Backstepping control

4.2.1  Backstepping for strict feedback systems

4.2.2  Adaptive backstepping control of electromechanical system

4.2.3  Adaptive backstepping control of T system

4.3  Periodic parametric perturbation control

4.3.1  Periodic parametric perturbation system

本书是在作者多年来对混沌系统理论的研究基础上编写而成.在简单介绍微分方程解析计算理论后,重点对平面微分方程的定性分析方法,包括流与流形的概念及计算,平面线性性系统的奇点分析,平面非线性系统线性化理论,平面非线性系统周期轨道的计算,判定,Poincare映射的计算,周期轨道的稳定性,及保守系统(Hamiltonian系统)和耗散系统的定性分析方法(能量函数法)进行介绍,同时对混沌理论的基本概念,以及混沌系统的Hopf分叉(中心流形的计算分析),混沌系统的降维(不变代数曲面的计算分析,无穷远点分析,奇异退化异宿环等),Melnikov方法对扰动Hamiltonian系统的分析等,和混沌系统的控制同步(包括自适应反步控制,周期参数扰动控制,
分数阶反馈控制,广义同步)理论进行介绍。