【作者】 朱玲；

【导师】 蒋威；

【作者基本信息】 安徽大学 ， 应用数学， 2007， 硕士

【摘要】 在经济、工程、生物等实际系统中，时滞现象和退化现象是非常普遍的现象。关于这两种现象，许多学者做出了很多的成果。但是，据我们所知，有许多系统同时具有退化和时滞的现象，称这类系统为退化时滞微分系统。也有许多系统具有多个时滞，称这类系统为多时滞系统。实际模型中某些参数的变化的变化会引起解的稳定性的变化，从而产生周期解或者(极限环)，即所谓的分支现象。这种现象在退化系统和生物系统中都很明显。本文主要研究三维退化时滞微分方程和多时滞捕食与食饵系统的稳定性与Hopf分支。我们以时滞τ做为分支参数研究了两类系统的hopf分支现象，通过分析系统的特征超越方程，发现当时滞穿越某些值的时出现了分支。结合利用Hopf分支定理获得了系统的Hopf分支存在的条件；利用中心流型定理和正规形方法分析了Hopf分支的性质，包括分支的方向和分支周期解的稳定性。第一章叙述了问题的产生背景与意义和本文所要做的工作。第二章讨论了三维退化时滞微分系统的Hopf分支存在性。第三章讨论了多时滞中立型捕食与食饵系统的Hopf分支存在性。第四章研究了多时滞捕食与食饵系统的Hopf分支的性质。更多还原

【Abstract】 In many practical fields, delay and degenerate phenomena are existing extensively, such as economy, engineering, biology and so on.About these phenomena many results have been obtained. But, as we know, delay and degenerate phenomena appear in many systems together, these systems are called degenerate differential system with delay. And some systems have two delay, we call these systems system with multiple delay.Some parameters’variation of models will cause the variation of stability of solutions, which can produce periodic solutions(or limit circle). Bifurcaion phenomenon is kind of it. This phenomenon is common in the degenerate differential system and ecolgical system This paper discusses the problems of Hopf bifurcation for three-neuron degenerate differential equations with delay and a neutral predator- prey model with two delay. We take delayτas a bifurcation parameter to investigate the hopf bifucation phenomena of thes two systems. By analyzing the associated characteristic transcendental equations of systems, it is found that Hopf bifurcation occurs when delay pass through a value, and using Hopf bifucation theorem, we obtain one condition for the existence of Hopf bifucation in systems. Furthermore, based on the center manifold theorem and the method of normal form, some results about properties of Hopf bifucation are obtained including the direction of Hopf bifucation and stablity of Hopf bifucating periodic solutions.In chapter one, the backgrounds and significance of the problems are given. The main work done in this paper is introduced.In chapter two, discusses the existence of Hopf bifurcation in three-neuron degenerate differential equations with delay. In chapter three, discusses the existence of Hopf bifurcation in a neutral predatorprdy model with two delay.In chapter four, studies the characters of Hopf bifucation.更多还原