状态依赖时滞微分方程的动力学研究

发布时间：2018-02-11 18:04

本文关键词： 状态依赖时滞 van der Pol模型捕食-被捕食者模型慢振荡解不动点定理喷射性摄动法超临界Hopf分岔次临界Hopf分岔　出处：《湖南大学》2016年博士论文　论文类型：学位论文

【摘要】：本文主要研究状态依赖时滞微分方程的动力学行为.关于状态依赖时滞微分方程目前还没有建立完善的理论体系,这一方面的理论和应用著作相对来说还很少.不过,随着很多学科如物理、化学、自动控制、人口增长、神经网络、细胞繁殖以及传染性疾病等领域的发展,越来越多的状态依赖时滞微分方程被用来描述一些模型的动力学行为.所以,对状态依赖时滞微分方程的深入研究具有非常重要的现实指导意义.本文主要研究了两类状态依赖时滞微分模型的动力学行为.主要内容如下:第一,我们研究了三种状态依赖时滞van der Pol模型的动力学行为.首先,我们通过分析第一种显性的时滞依赖于状态的van der Pol方程的结构,得到了解的有界性、存在唯一性、以及解对初值的连续依赖性;通过构造一个适当的紧集和一个连续的紧映射,再根据其平衡态的喷射性,运用不动点定理得到其慢振荡周期解的存在性.然后,我们研究了第二种由状态和时滞一起决定的微分形式的时滞van der Pol模型的慢振荡周期解问题.这个模型和前一个van der Pol模型的区别在于其时滞函数的变化.这种时滞的变化,使得所构造的后继映射对于时滞项在平衡态处是不连续的.为了能够利用合适的不动点定理,我们通过寻找适当的紧集,并且在其上进行对时间项的单位化,以此来构造一个辅助的紧状态空间,再在这个状态空间上构造后继映射,并为了解决时滞项在平衡态的不连续问题,我们构造一个后继映射的辅助映射,从而使得我们在其上构造的系统在辅助紧状态空间中保持不变,利用拟喷射不动点研究局部动力学性质,证明了其慢振荡周期解的存在性.最后,我们详细讨论了第三种状态依赖时滞微分van der Pol方程的动力学行为:我们研究了其平衡点的局部稳定性,解的渐近性,以及其Hopf分岔的存在性;通过摄动过程以及Fredholm正交公式得到了方程的局部分岔是超临界Hopf分岔还是次临界Hopf分岔的判定,并得到其分岔出的周期解的稳定性;利用三种常见的时滞函数,用数值模拟的方法,验证了我们所提供的理论分析.第二,我们研究了一类捕食-被捕食者过程模型的动力学行为.首先,在对方程中的几个参数作出适当假设的情况下,我们研究了模型中正平衡点的局部稳定性、解的振荡性等动力学性质,寻找到了解产生Hopf分岔的参数范围;通过将原系统在正平衡点处摄动化,把状态依赖时滞微分方程的Hopf分岔问题转化为常时滞微分方程的Hopf分岔问题;通过Fredholm正交公式得到了方程的局部Hopf分岔定理,从而得到其是超临界Hopf分岔或次临界Hopf分岔的判定以及分岔所产生的周期解的稳定性;利用三种常见的时滞函数,用数值模拟的方法,验证了我们所得到的理论分析.
[Abstract]:In this paper, we mainly study the dynamical behavior of state dependent delay differential equations. There is no perfect theoretical system for state dependent delay differential equations at present, but there are few theoretical and practical works on this aspect. With the development of many fields such as physics, chemistry, automatic control, population growth, neural network, cell reproduction and infectious diseases, More and more state-dependent delay differential equations are used to describe the dynamic behavior of some models. In this paper, the dynamic behavior of two classes of state-dependent delay differential models is studied. The main contents are as follows: first, We study the dynamical behavior of three state-dependent delay van der Pol models. Firstly, we obtain the boundedness and uniqueness of the solution by analyzing the structure of the first explicit delay-dependent van der Pol equation. By constructing a proper compact set and a continuous compact mapping, and according to the ejection of its equilibrium state, the existence of periodic solutions of its slow oscillation is obtained by using the fixed point theorem. In this paper, we study the slow oscillatory periodic solution of the second kind of delay van der Pol model, which is determined by both state and delay. The difference between this model and the previous van der Pol model lies in the variation of the delay function. In order to make use of the suitable fixed point theorem, we find the appropriate compact set and make the time term unit on it. In order to solve the discontinuity problem of the delay term in the equilibrium state, we construct an auxiliary mapping of the successor mapping, in order to construct an auxiliary compact state space, and then construct a successor map on the state space, in order to solve the discontinuity problem of the delay term in the equilibrium state, we construct an auxiliary mapping of the successor mapping. Therefore, the system constructed on it remains invariant in the auxiliary compact state space. The local dynamical properties of the system are studied by using quasi-jet fixed points, and the existence of periodic solutions for its slow oscillation is proved. We discuss the dynamic behavior of the third state-dependent delay differential van der Pol equation in detail. We study the local stability of the equilibrium point, the asymptotic behavior of the solution and the existence of its Hopf bifurcation. By means of perturbation process and Fredholm orthogonal formula, the local bifurcation of the equation is determined by supercritical Hopf bifurcation or subcritical Hopf bifurcation, and the stability of the periodic solution of the bifurcation is obtained. The numerical simulation method is used to verify the theoretical analysis provided by us. Secondly, we study the dynamic behavior of a kind of predator-prey process model. In this paper, we study the local stability and oscillatory properties of the positive equilibrium point in the model, and find out the parameter range of the Hopf bifurcation, by perturbing the original system at the positive equilibrium point. The Hopf bifurcation problem of the state dependent delay differential equation is transformed into the Hopf bifurcation problem of the ordinary delay differential equation, and the local Hopf bifurcation theorem of the equation is obtained by the Fredholm orthogonal formula. The results show that the bifurcation is a supercritical Hopf bifurcation or a subcritical Hopf bifurcation and the stability of the periodic solution generated by the bifurcation is verified by using three common time-delay functions and the numerical simulation method.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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