Preface

Chapter 1 A Brief Description
1. Linear Differential Equations
2. The Need for Qualitative Analysis
3. Description and Terminology

Chapter 2 Existence and Uniqueness
1. Introduction
2. Existence and Uniqueness
3. Dependence on Initial Data and Parameters
4. Maximal Interval of Existence
5. Fixed Point Method

Chapter 3 Linear Differential Equations
1. Introduction
2. General Nonhomogeneous Linear Equations
3. Linear Equations with Constant Coefficients
4. Periodic Coefficients and Floquet Theory

Chapter 4 Autonomous Differential Equations in R2
1. Introduction
2. Linear Autonomous Equations in R2
3. Perturbations on Linear Equations in R2
4. An Application： A Simple Pendulum

Chapter 5 Stability
1. Introduction
2. Linear Differential Equations
3. Perturbations on Linear Equations
4. Liapunovs Method for Autonomous Equations

Chapter 6 Periodic Solutions
1. Introduction
2. Linear Differential Equations
3. Nonlinear Differential Equations

Chapter 7 Dynamical Systems
1. Introduction
2. Poincare-Bendixson Theorem in R2
3. Limit Cycles
4. An Application： Lotka-Volterra Equation

Chapter 8 Some New Equations
1. Introduction
2. Finite Delay Differential Equations
3. Infinite Delay Differential Equations
4. Integrodifferential Equations
5. Impulsive Differential Equations
6. Equations with Nonlocal Conditions
7. Impulsive Equations with Nonlocal Conditions
8. Abstract Differential Equations
Appendix
References
Index

精彩书摘

The study of linear differential equations is very important for the fol-lowing reasons. First， the study provides us with some basic knowledgefor understanding general nonlinear differential equations. Second， manynonlinear differential equations can be written as summations of linear dif-ferential equations and some small nonlinear perturbations. Thus， undercertain conditions， the qualitative properties of linear differential equationscan be used to infer essentially the same qualitative properties for nonlineardifferential equations.

前言/序言

　　Differential equations are mainly used to describe the changes of quanti-ties or behavior of certain systems in applications， such as those governedby Newtons laws in physics.
　　When the differential equations under study are linear， the conventionalmethods， such as the Laplace transform method and the power series solu-tions， can be used to solve the differential equations analytically， that is， thesolutions can be written out using formulas.
　　When the differential equations under study are nonlinear， analytical so-lutions cannot， in general， be found; that is， solutions cannot be writtenout using formulas. In those cases， one approach is to use numerical ap-proximations. In fact， the recent advances in computer technology makethe numerical approximation classes very popular because powerful softwareallows students to quickly approximate solutions of nonlinear differentialequations and visualize their properties.
　　However， in most applications in biology， chemistry， and physics mod-eled by nonlinear differential equations where analytical solutions may beunavailable， people are interested in the questions related to the so-calledqualitative properties， such as： will the system have at least one solu-tion？ will the system have at most one solution？ can certain behavior ofthe system be controlled or stabilized？ or will the system exhibit some peri-odicity？ If these questions can be answered without solving the differentialequations， especially when analytical solutions are unavailable， we can stillget a very good understanding of the system. Therefore， besides learningsome numerical methods， it is also important and beneficial to learn howto analyze some qualitative properties.

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