两类动力系统的稳定性与分岔分析

发布时间：2018-01-15 09:18

本文关键词：两类动力系统的稳定性与分岔分析　出处：《郑州大学》2015年硕士论文　论文类型：学位论文

【摘要】：动力系统是非线性科学的重要组成部分,是现代数学的重要分支之一,混沌和分岔是非线性动力系统中新的存在形式且是重要的研究领域,对其研究的成果可广泛应用于物理、力学、化学、生物及经济学等具有更强实用性的研究领域.本文主要研究了一类猝变动力及一类传染病模型的动力特性,分为如下四个章节进行讨论：第一章介绍了本文的研究背景及研究现状；第二章介绍了研究模型动力性质的相关预备知识;第三章研究了一类猝变动力的混沌及Hopf分岔现象,分析了其平衡点的稳定性、Hopf分岔的存在性,运用规范型定理分析了其分岔周期解的方向及稳定性,最后,通过数值模拟验证了理论分析的结果,并发现了通过其倍周期分岔可以引起一个费根鲍姆类型的奇怪吸引子；第四章研究了一类具有偏永久免疫的儿童传染病模型,计算了其基本再生数,分析了其无病平衡点及地方病平衡点的局部稳定性和全局稳定性,并分析了平衡点处的分岔现象,结果显示,当治愈率τ超过一临界值τ*时,在地方病平衡点处会分岔出周期轨,最后给出数值例子模拟,模拟结果与理论分析结果相吻合.
[Abstract]:Dynamic system is an important part of nonlinear science and one of the important branches of modern mathematics. Chaos and bifurcation are new forms and important research fields in nonlinear dynamic system. The research results can be widely used in the fields of physics, mechanics, chemistry, biology and economics, etc. In this paper, the dynamic characteristics of a kind of quenching force and a class of infectious disease models are studied. It is divided into the following four chapters to discuss: the first chapter introduces the research background and research status of this paper; The second chapter introduces the related preparatory knowledge of the dynamic properties of the model. In chapter 3, we study the chaos and Hopf bifurcation of a class of quenched dynamics, and analyze the existence of the stability of the equilibrium point and the Hopf bifurcation. The direction and stability of the bifurcation periodic solution are analyzed by using the normal form theorem. Finally, the results of the theoretical analysis are verified by numerical simulation. It is also found that a Fegenbaum type strange attractor can be induced by its double period bifurcation. In chapter 4th, we study a class of infective disease models with partial permanent immunity, calculate their basic regeneration numbers, and analyze the local stability and global stability of disease-free equilibrium and endemic equilibrium. The bifurcation phenomenon at the equilibrium point is analyzed. The results show that when the cure rate 蟿 exceeds a critical value 蟿 *, there will be a periodic orbit at the endemic equilibrium point. Finally, a numerical example is given. The simulation results are in agreement with the theoretical analysis results.
【学位授予单位】：郑州大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

【共引文献】