Global Dynamics Analysis Of Several Classes Of Delay Differential Equation

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.