## Maple常微分方程稳定性及其应用

Title  Ordinary Differential Equations stability theory  and its application
Abstract
Ordinary differential equation has important applications in many aspects such as power system. As in many practical problems of ordinary differentia -l equations is difficult to find explicit solutions, so we must through the equation itself to determine the stability of the solution.
This paper first introduces the basic theory of ordinary differential equations, which is the foundation of stability theory, and then the stability of solutions of ordinary differential equations, which is the Lyapunov stability. Lyapunov stability refers to when time tends to infinity, continuous dependence of solutions on initial values. Then through the Maple programming, the phase diagram of differential equation, the change of properties of solutions. Finally, the stability of the application in practice.
Keywords  Ordinary Differential Equations   Stability   Null Solution
Lyapunov Function

1 引言    1
1.1  研究背景及意义    1
1.2  研究现状    2 源自六@维\$论`文^网"加7位QQ3249`114 www.lwfree.cn
1.3  本文的主要内容    2
2  常微分方程基本定理    3
2.1 解的存在唯一性定理.    3
2.2  解的延拓性定理.    4
2.3  解对初值和参数的连续依赖性.    5
3  常微分方程的稳定性    7
3.1  稳定性的概念.    7
3.2  自治系统解的稳定性.    10
3.2.1  李雅普诺夫函数.    10
3.2.2  稳定性定理.    12
3.2.3  不稳定性定理.    14
3.3  非自治系统解的稳定性.    15
3.3.1  李雅普诺夫函数和 类函数.    15
3.3.2  稳定性定理.    17
3.3.3不稳定性定理.    19
3.4  全局稳定性.    20
4  李雅普诺夫函数的构造    28
4.1  李雅普诺夫函数的存在性    28
4.2    李雅普诺夫函数构造方法    29
4.2.1  巴尔巴欣公式    30
4.2.2  线性类比法    31
4.2.3  能量函数法    32
4.2.4  分离变量法    33
4.2.5  变梯度法    34
5  常微分方程稳定性应用举例    36
5.1    综合国力的微分方程模型的稳定性分析    36
5.2    产品市场调节的稳定性分析    38

1 引言

------分隔线----------------------------