非单调反馈时滞微分方程的多重稳定性分析

发布时间：2018-08-23 09:46

【摘要】：本文研究的主要目的是构建一种分析标量时滞微分方程(?)(t)=-g(x(t))+f(x(t-τ))的多重稳定性动力学行为的方法。在已有的区域分解法、单调动力系统和一维映射分析方法的基础上,对半流的正不变区域和吸引盆进行了估计。同时,详细证明了Allee-Type模型的全局动力学行为。具体而言,证明了Hopf分支和异宿轨的存在性(包括两种异宿轨:一种是从一个平衡点到另一个平衡点,另一种是从一个平衡点到最大正平衡点周围的周期轨)。最后,给出了数值模拟的例子来验证本文的一般性结论。全文由四部分组成。第一章,对时滞系统进行了简要的介绍,然后对本文所研究的模型产生的背景以及我们研究的主要内容进行概述。第二章,我们给出了基本的定义和背景理论的解释。第三章,研究模型(?)(t)=-g(x(t))+f(x(t-τ))在不变区间上平衡点的多重稳定性质。第四章,我们详细研究了Allee-Type模型的多重稳定性质。特别地是,证明了从一个平衡点到另一个平衡点或到最大平衡点周围的周期轨的存在性。同时,给出数值模拟的例子来阐述本文的主要结论。
[Abstract]:The main purpose of this paper is to construct a method to analyze the multi-stability dynamic behavior of scalar delay differential equations (?) (t)? -g (x (t) f (x (t- 蟿). Based on the domain decomposition method, monotone dynamical system and one-dimensional mapping analysis method, the positive invariant region and the attraction basin of half-flow are estimated. At the same time, the global dynamic behavior of Allee-Type model is proved in detail. In particular, the existence of Hopf bifurcation and heteroclinic orbit is proved (including two kinds of heteroclinic orbits: one is from one equilibrium point to the other equilibrium point, the other is from one equilibrium point to the periodic orbit around the maximum positive equilibrium point). Finally, an example of numerical simulation is given to verify the general conclusion of this paper. The full text consists of four parts. In the first chapter, we briefly introduce the time-delay system, and then summarize the background of the model and the main contents of our research. In the second chapter, we give the basic definition and the explanation of background theory. In chapter 3, we study the multiple stability properties of equilibrium points on invariant intervals of the model (?) (t) + -g (x (t) f (x (t- 蟿). In chapter 4, we study the multiple stability of Allee-Type model in detail. In particular, the existence of periodic orbits from one equilibrium point to another equilibrium point or to the maximum equilibrium point is proved. At the same time, an example of numerical simulation is given to illustrate the main conclusions of this paper.
【学位授予单位】：长沙理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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