小时滞梯度系统的动力学行为

数学物理学报 ›› 2015, Vol. 35 ›› Issue (3): 464-477.

小时滞梯度系统的动力学行为

尹逊武¹, 李德生²

1. 天津工业大学理学院天津 300387;
2. 天津大学理学院天津 300072

收稿日期:2014-05-06 修回日期:2015-03-25 出版日期:2015-06-25 发布日期:2015-06-25
作者简介:尹逊武, yinxunwu@tjpu.edu.cn;李德生, lidsmath@tju.edu.cn
基金资助:
国家自然科学基金 (11471240, 11071185)和天津市高等学校科技发展基金计划项目(20141002)资助

Dynamical Behavior of Gradient System with Small Time Delay

Yin Xunwu¹, Li Desheng²

1. School of Science, Tianjin Polytechnic University, Tianjin 300387;
2. School of Science, Tianjin University, Tianjin 300072

Received:2014-05-06 Revised:2015-03-25 Online:2015-06-25 Published:2015-06-25

摘要/Abstract

摘要：

该文研究下列具有小时滞的一般非线性梯度型发展方程 ∂_tu+Au=f(u(t),u(t-τ)). 证明了当时间趋于无穷大时, 时滞方程的每一个有界解将收敛于某一个平衡点, 只要时滞足够小, 这意味着时滞系统的行为非常类似非时滞系统. 这里的方法主要是基于梯度系统不变集的Morse结构和发展方程的几何理论. 这个结果的证明分两步完成: 首先, 在梯度系统和有限个孤立平衡点的假设下, 证明了一定存在一个足够小的时滞使得时滞方程的任一个有界解将会最终进入并停留在某一个平衡点的邻域里面; 其次, 在双曲平衡点的假设下, 运用指数二分性和一系列的估计, 证明了一定存在ε>0 和足够小的 τ>0使得任一个落于某个平衡点ε -邻域内的解最终收敛于该平衡点, 当时间趋于无穷大时.

关键词: 孤立双曲平衡点, Aubin-Lions 引理, 梯度系统, Morse结构, 指数二分性

Abstract:

In this article, we investigate the dynamical behavior of the following general nonlinear gradient-like evolutionary equation with small time delay ∂_tu+Au=f(u(t),u(t-τ)). We prove that each bounded solution of the delayed equation will converge to some equilibrium as t→∞ provided the delay is sufficiently small. This indicates that gradient system with small time delay behaves very much like the nondelayed one. The approach here is mainly based on the Morse structure of invariant sets of gradient system and some geometric analysis of evolutionary equations. The proof of this result is completed in two steps. First, with the hypothesis of gradient system, finite and isolated equilibria, we prove that there exists a sufficiently small delay such that any bounded solution of the delayed equation will ultimately enter and stay in the neighborhood of one equilibrium. Second, with the hypothesis of hyperbolic equilibrium, we utilize exponential dichotomies and a series estimates to prove that there exists ε > 0 and τ > 0 sufficiently small such that any solution of the delayed equation lying in the ε-neighborhood of one equilibrium will converge to this equilibrium as t → ∞.

Key words: Isolated hyperbolic equilibrium, Aubin-Lions lemma, Gradient system, Morse structure, Exponential dichotomies

中图分类号:

O193.4

尹逊武, 李德生. 小时滞梯度系统的动力学行为[J]. 数学物理学报, 2015, 35(3): 464-477.

Yin Xunwu, Li Desheng. Dynamical Behavior of Gradient System with Small Time Delay[J]. Acta mathematica scientia,Series A, 2015, 35(3): 464-477.

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