【作者】 阳超；

【导师】 黄立宏；

【作者基本信息】 湖南大学 ， 应用数学， 2017， 博士

【摘要】 在现实生活中的各个领域里,不连续自然现象随处可见,在不连续环境下的讨论模型是为了使数学模型有更好的应用背景,并能应用到更实际的领域中.故研究右端不连续的数学模型逐渐成为当今众多学者所研究的热点问题之一.特别是在神经网络模型和具有不连续捕获的生物种群模型中,右端不连续微分方程更是得到前所未有的推广.而如今,对神经网络模型的研究方式也不再单一,由于各个神经元内部之间的相互联系,相互影响,本文考虑了带耦合和切换的复杂神经网络系统同步稳定化控制等问题.此外,考虑到生态种群的环境多样化,时滞与伪概周期变化性环境对种群的影响,本文还分别研究了在时滞周期环境和时滞伪概周期环境下的生物种群模型.主要包含三部分内容:右端不连续微分方程的同步稳定化控制问题;右端不连续种群模型的动力学问题;带分布时滞的生物种群的伪概周期问题.具体的研究工作如下:在第一章中,简单回顾了本文研究的背景和现状,并且对本文的具体的工作进行简要的概括,同时研究的具体的应用方向和学习动机也进行扼要的叙述.然后提出本文的结构安排和主要内容.在第二章中,给出一些基本知识和基本定义,以及本文所用到的一些基本引理与方法.主要是右端不连续方程,非光滑分析,矩阵理论等内容.在第三章中,利用微分包含和集值分析理论,通过构造Lyapunov函数的方法.通过设计几类新颖的控制器,分别研究了具有耦合时滞自治和非自治右端不连续复杂神经网络系统的有限时间同步和全局指数同步问题.同时,针对右端不连续的多元函数的同步性,也得到了其全局指数稳定的结果.最后通过数值例子来验证本章结果的有效性.在第四章中,研究了一类具有不连续捕获项的混合时滞Lasota-Wazewska模型,其处环境为非光滑的周期性变化动力环境.基于非光滑分析,不动点理论以及Lyapunov方法,得到了该模型周期解的存在性和全局指数稳定性结论.最后通过数值例子来验证本章结果的有效性.在第五章中,研究了一类带分布时滞的Nicholson飞蝇方程模型.通过伪概周期函数理论,不动点方法.设计一个恰当的Lyapunov函数,最后得到了系统伪概周期解存在性和渐近稳定性以及全局指数渐近稳定性结论.更多还原

【Abstract】 Discontinuous natural phenomena can be seen in many fields of real world.And the mathematical models what we generally studied are to apply to the actual operation more directly.As the assumed condition of the traditional continuous mathematical model is single and the application background and the actual ap-plication field are limited,the right-hand discontinuous mathematical model has gradually become one of hot issues and studied by many researchers.Especially in the models of the complex neural network and the biological population with discontinuous harvesting,the right-hand discontinuous differential equations are promoted unprecedentedly.Now,the research on complex neural network model is no longer single,due to the interconnection of each neuron and mutual influ-ence.Meanwhile,a series of problems such as synchronization stabilizing control on complex neural network system with coupling and switching must be taken into consideration.On the other hand,considering the influence of the periodic envi-ronmental of ecological population,pseudo almost periodic variability environment and time-varying delayed to population.This paper studies the biological popu-lation model under the time-varying delayed periodic and pseudo almost periodic environmental conditions.It consists of three parts:the synchronization stabiliz-ing control problem of discontinuous differential equation;the dynamics problem of right-hand discontinuous population model;the pseudo almost periodic problem of the biological population with time delayed.The main research is as follows:In Chapter 1,A brief review of the background and current situation of this study,we summarize the specific work,and the application of the specific direction and motivation are briefly described.Then the structure and main contents of this paper are presented.In Chapter 2,Some basic knowledge and basic definitions are given,as well as some basic lemmas and methods used in this paper,such as the right-hand discontinuous equations,nonsmooth analysis,matrix theory and so on.In Chapter 3,based on differential inclusions and the theory of set-valued anal-ysis method,by constructing a Lyapunov function.Through the design of several kinds of novel controller,we study autonomous and nonautonomous time-delayed coupled complex discontinuous neural networks,and the finite-time synchroniza-tion and global exponential synchronization are verified,respectively.Meanwhile,we also gain the global exponential synchronization criteria for multiple function with right-hand discontinuous sides.Finally,we present an example with simula-tion to illustrate our theoretical results.In Chapter 4,we study a class of mixed time-varying delayed Lasota-Wazewska model with discontinuous harvesting,which is described by a periodic nonsmooth dynamical system.Base on nonsmooth analysis,Kakutani’s fixed point method and the generalized Lyapunov method,easily verifiable delay-independent criteria are established to ensure the existence and exponential stability of positive peri-odic solutions.Finally,we present an example with simulation to illustrate the theoretical results.In Chapter 5,we study a class of Nicholson’s blowflies model with distribut-ed delays.Base on properties of pseudo almost periodic functions,fixed point method and the generalized Lyapunov method,new criteria are established to en-sure the existence and asymptotic stability of pseudo almost periodic solutions,which cover and complement some existing ones.Finally,we present an example with simulation to illustrate the theoretical results.更多还原