非线性系统的研究近年来受到越来越广泛的关注，国外许多工科院校已将“非线性系统”作为相关专业研究生的学位课程。本书介绍了非线性系统的基本概念和基本分析方法；输入-输出稳定性、无源性和反馈系统的频域分析；现代稳定性分析的基本概念、扰动系统的稳定性、扰动理论和平均化以及奇异扰动理论；反馈控制的基本概念的反馈线性化，并给出了几种非线性设计工具，如滑模控制、Lyapunov再设计、反步法、基于无源的控制和高增益观测器等。此外本书附录还汇集了一些书中用到的数学知识，包括基本数学知识的复习、压缩映射和一些较为复杂的定理证明。

本书既可以作为研究生第一学期非线性系统课程的教材，也可以作为工程技术人员、应用数学专业人员的自学教材或参考书。

Preface

Acknowledgments

Standard Notation

1 Linear vs. Nonlinear

　1.1 Nonlinear Models

　1.2 Complexity in Nonlinear Dynamics

　　1.2.1 Subtleties of Nonlinear Systems Analysis

　　1.2.2 Autonomous Systems and Equilibrium Points

　1.3 Some Classical Examples

　　1.3.1 The Tunnel Diode Circuit

　　1.3.2 An Oscillating Circuit: Due to van der Po!

　　1.3.3 The Pendulum: Due to Newton

　　1.3.4 The Buckling Beam: Due to Euler

　　1.3.5 The Volterra-Lotka Predator-Prey Equations

　1.4 Other Classics: Musical Instruments

　　1.4.1 Blowing of a Clarinet Reed: Due to Rayleigh

　　1.4.2 Bowing of a Violin String: Due to Rayleigh

　1.5 Summary

　1.6 Exercises

2 Planar Dynamical Systems

　2.1 Introduction

　2.2 Linearization About Equilibria of Second-Order Nonlinear Systems

　　2.2.1 Linear Systems in the Plane

　　2.2.2 Phase Portraits near Hyperbolic Equilibria

　2.3 Closed Orbits of Planar Dynamical Systems

　2.4 Counting Equilibria: Index Theory

　2.5 Bifurcations

　2.6 Bifurcation Study of Josephson Junction Equations

　2.7 The Degenerate van der Pol Equation

　2.8 Planar Discrete-Time Systems

　　2.8.1 Fixed Points and the Hartman-Grobman Theorem

　　2.8.2 Period N Points of Maps

　　2.8.3 Bifurcations of Maps

　2.9 Summary

　2.10 Exercises

3 Mathematical Background

　3.1 Groups and Fields

　3.2 Vector Spaces, Algebras, Norms, and Induced Norms

　3.3 Contraction Mapping Theorems

　　3.3.1 Incremental Small Gain Theorem

　3.4 Existence and Uniqueness Theorems for Ordinary Differential Equations

　　3.4.1 Dependence on Initial Conditions on Infinite Time Intervals

　　3.4.2 Circuit Simulation by Waveform Relaxation

　3.5 Differential Equations with Discontinuities

　3.6 Carleman Linearization

　3.7 Degree Theory

　3.8 Degree Theory and Solutions of Resistive Networks

　3.9 Basics of Differential Topology

　　3.9.1 Smooth Manifolds and Smooth Maps

　　3.9.2 Tangent Spaces and Derivatives

　　3.9.3 Regular Values

　　3.9.4 Manifolds with Boundary

　3.10 Summary

　3.11 Exercises

4 Input-Output Analysis

5 Lyapunov Stability THeory

6 Applications of Lyapunov THeory

7 Dynamical Systems and Bifurcations

8 Basics of Differential Geometry

9 Linearization by State Feedback

10 Design Examples Using Linearization

11 Geometric Nonlinear Control

12 Exterior Differential Systems in Control

13 New Vistas: Multi-Agent Hybrid Systems

References

Index