几类Filippov系统与光滑微分系统的定性理论及应用研究

发布时间：2018-04-12 19:34

本文选题：右端不连续微分方程 + Σ-奇异点　；参考：《湖南大学》2015年博士论文

【摘要】：由于数学分析工具的缺乏,右端不连续微分方程理论的发展相当缓慢.至今,右端不连续微分方程理论的发展仍处于初级阶段,很多理论都没有得到完善.为此,本学位论文首先发展了有关右端不连续非自治广义齐次微分方程的稳定性理论,主要讨论了右端不连续非自治广义齐次微分方程以及一类具有不连续扰动项的齐次系统解的收敛方式.然后,我们根据现实生活中一些食饵和捕食者之间的接触特征,建立了一个具有不连续功能反应函数的食饵捕食模型,并应用右端不连续微分方程定性理论对该模型进行了深入的研究.在本学位论文的最后两部分,我们通过构造Lyapunov函数的方法分别讨论了一类光滑的HIV双仓室模型和一类具有不连续激励函数的神经网络模型平衡态的稳定性.值得一提的是,在研究具有不连续激励函数的神经网络模型平衡态的稳定性时,我们获得了在一定的条件下,平衡态是有限时间收敛的.该收敛方式在光滑的HIV仓室模型中是不可能具有的.具体地说,本学位论文主要分为六章. 在第一章中,我们首先回顾了右端不连续微分方程和稳定性理论的研究历史和发展概况.随后,我们逐一地介绍了不连续生物动力学、HIV动力学和不连续神经网络动力学的研究历史和发展现状. 在第二章中,我们简要地介绍了本学位论文所需的一些数学理论知识,主要包括Filippov系统的定性理论,如解的存在唯一性、解对初值和右端函数的依赖性、平面Filippov系统中不连续曲面上解的特性、∑-奇异点附近解的拓扑结构.同时, 一些有关光滑动力系统和Filippov系统的稳定性理论也在此处被列出. 在第三章中,我们完善了右端不连续非自治广义齐次微分方程的稳定性理论.利用齐次微分方程的特性(收缩解仍是原方程的解)和一些运算上的处理技巧,得到了右端不连续非自治广义齐次微分方程解的收敛特征：齐次度为零和正的非自治广义齐次系统,全局渐近稳定的平衡点分别是指数稳定和1/t型稳定的.同时,我们分析了一类具有内外扰动项的扰动系统的动力学行为,并将其与原系统进行了对比研究.结果显示该扰动系统与原系统具有类似的动力学行为,当扰动充分小时,解的收敛方式在该扰动下是稳定的. 在第四章中,我们根据在现实生活中一些具有比率依赖的食饵与捕食者之间的接触特性,建立了一个比率依赖的Filippov食饵捕食模型.主要运用右端不连续微分方程定性理论中有关∑-奇异点拓扑结构的相关知识,对该模型所有∑-奇异点的局部结构进行了详细的分析.此外,对于不同的参数,我们得到了该模型具有14种不同的全局动力学行为.特别地,对于一定范围内的参数,全局稳定的伪平衡点和全局有限时间稳定的周期解是存在的,并且所得的部分动力学行为能合理地解释部分实验和现实中的现象. 在第五章中,我们将构造Lyapunov函数的方法与LaSalle不变原理相结合,分别讨论了一类光滑的七维和八维HIV仓室模型平衡态的渐近稳定性.通过对这两个模型的分析,我们得到了:当基本再生数小于等于1时(即感染平衡点不存在),无病平衡点是全局渐近稳定的;当基本再生数大于1时(即感染平衡点存在),感染平衡点是全局渐近稳定的. 在第六章中,根据周期解是否与不连续曲面相切,我们研究了一类具有不连续激励函数的Hopfeld神经网络模型周期解的有限时间收敛性.当周期解和不连续曲面相切时,周期解一定是有限时间稳定的.当周期解和所有不连续曲面横截相交时,在一定的条件下,通过构造Lyapunov函数法可得周期解的有限时间收敛性.
[Abstract]:Due to the lack of tools of mathematical analysis, discontinuous development theory of differential equations is very slow. So far, the right end is not continuous development of the theory of differential equation is still in the primary stage, many theories are not perfect. Therefore, this thesis firstly developed the discontinuous nonautonomous generalized homogeneous differential equation stability the theory, mainly discusses the discontinuous nonautonomous generalized homogeneous differential equations and a class of discontinuous solutions of the perturbation convergence mode of homogeneous system. Then, we according to the contact characteristics between the real life of some prey and predator, has not established a continuous functional response predator-prey model and application of discontinuous qualitative theory of differential equations of the model are studied. In the last two parts of this thesis, we construct the Lyapunov function. Method are used to discuss the stability of a class of smooth HIV double compartment model and a neural network model of equilibrium discontinuous activation function. It is worth mentioning that the stability of neural networks model with equilibrium discontinuous activation functions in the research, we obtained under certain conditions, the equilibrium state finite time convergence. The convergence is not possible with the HIV chamber smooth model. Specifically, this thesis is mainly divided into six chapters.
In the first chapter, we first review the research history and development of the right end discontinuous differential equations and stability theory. Next, we introduce the history and development of discontinuous biologic dynamics, HIV dynamics and discontinuous neural network dynamics.
In the second chapter, we briefly introduce some mathematical theory in this thesis is required, including the qualitative theory of Filippov system, such as the existence and uniqueness of solutions, solutions to the initial value and the right function dependent characteristics of planar Filippov system of discontinuous surface solution, solution of the topological structure near sigma singular points. At the same time,
Some stability theories about smooth dynamic systems and Filippov systems are also listed here.
In the third chapter, we improve the discontinuous nonautonomous generalized homogeneous differential equation stability theory. Based on the characteristics of homogeneous differential equations (contraction of solution is the solution of the equation) and some computing skills, the discontinuous non convergence of generalized homogeneous differential equations autonomy: homogeneous zero and positive non autonomous generalized homogeneous system, global asymptotic stability of the equilibrium point are exponential stability and 1/t stability. At the same time, we analyze the dynamic behavior of a class of perturbed system of internal and external disturbances, which were compared with the original system. The results showed the disturbance of system and the original system has similar dynamical behavior, when the perturbation is sufficiently small, the convergence solutions is stable in the disturbance.
In the fourth chapter, according to the US in real life has some contact characteristics between prey and predator ratio dependent, established Filippov predator-prey model with a ratio dependent. Mainly using the discontinuous of sigma in qualitative theory of differential equation of singular point related knowledge of topology, the local structure of the the model of all sigma singular points are analyzed in detail. In addition, for different parameters, we get the model with 14 different global dynamics. In particular, the parameters for a certain range, the global stability of the equilibrium point and periodic pseudo global finite time stable solutions exist, dynamics the behavior and can reasonably explain some experiments and the reality of the phenomenon.
In the fifth chapter, we will construct the Lyapunov function method and LaSalle invariance principle combined respectively asymptotic stability of a class of smooth seven and eight dimensional HIV compartment model of equilibrium is discussed. Through the analysis of these two models, we obtained: when the basic reproduction number is smaller than or equal to 1 (i.e. the infected equilibrium does not exist), the disease-free equilibrium is globally asymptotically stable; when the basic reproduction number is greater than 1 (i.e. the infection equilibrium exists), the infection free equilibrium is globally asymptotically stable.
In the sixth chapter, according to the periodic solution and discontinuous tangent to the surface, we study a class of finite time convergence of periodic Hopfeld neural network model of discontinuous solution. When the excitation function of periodic solutions and discontinuous surface tangent, periodic solutions are finite time stable. When the periodic solutions and all discontinuities surface transverse intersection, under certain conditions, by constructing Lyapunov function method can converge in finite time of periodic solutions.

【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

【参考文献】