常微分方程平衡点及稳定性研究38112

摘要

本文给出了微分方程稳定性的概念,并举了一些例子来说明不同稳定性定义之间的区别和联系。这些例子都是通过求出方程解析解的方法来讨论零解是否稳定。在实际问题中提出的微分方程往往是很复杂的,无法求出其解析解,这就需要我们从方程本身来判断零解的稳定性。所以我们讨论了通过Liapunov稳定性定理来判断自治系统零解的稳定性，并用类似的方法讨论了非自治系统零解的稳定性。在此基础上，讨论了一阶和二阶微分方程的平衡点及其稳定性，这对其研究数学建模的稳定性模型起到很大的作用,并且利用相关的差分方程的全局吸引性研究了具时滞的单种群模型

()()()() ()

.1

1N t

N t r t N t

cN t ττ

--

=

--

的平衡点1

x=的全局吸引性,所获结果改进了文献中相关的结论。关键词：自治系统平衡点稳定性全局吸引性

Abstract

In this paper,we gived the conceptions of differential equation stability. Simultaneously a number of examples to illustrate the difference between the definition of different stability and contact. These examples are obtained by analytical solution equation method to discuss the stability of zero solution. Practical issues raised in the often very complicated differential equations, analytical solution can not be obtained, which requires us to determine from the equation itself, the stability of zero solution. So we discussed the stability theorem to determine through the stability of zero solution of autonomous systems, and use similar methods to discuss the non-zero solution of autonomous system stability. On this basis,we discuss a step and the second-step and the stability, which plays the major role to its stability of the model, and the global attractivity of the positive equilibrium 1

x=of the following delay single population model

()()()() ()

.1

1N t

N t r t N t

cN t ττ

--

=

--

is investigated by using the corresponding result related to a difference equation.The obtained results improve some known results in the literature.

Key Words:autonomous system;equilibrium point;stability;delay;globally asymptotic stability;global attractivity

目录

摘要...............................................................................................................................I Abstract ......................................................................................................................... II 目录...............................................................................................................................I 第1章引言 (1)

第2章微分方程平衡点及稳定性分析 (3)

2.1 平衡点及稳定性定义 (3)

2.2 自治系统零解的稳定性 (4)

2.2.1 V函数 (4)

2.2.2 Liapunov稳定性定理 (5)

2.3 非自治系统的稳定性 (8)

2.3.1 V函数和k类函数 (8)

2.3.2 零解的稳定性 (10)

2.4 判定一阶微分方程平衡点稳定性的方法 (14)

2.4.1 相关定义 (14)

2.4.2 判定平衡点稳定性的方法 (14)

2.5 判定二阶微分方程平衡点稳定性的方法 (15)

2.5.1 相关定义 (15)

2.5.2 判定平衡点稳定性的方法 (15)

第3章一类时滞微分方程平衡点的全局吸引性 (17)

3.1 差分方程(3-7)的全局渐近稳定性 (17)

3.2 微分方程(3-1)的全局吸引性 (19)

第4章常微分方程稳定性的一个应用 (23)

第5章结论 (25)

参考文献 (27)

致谢 (29)