Stability And Hopf Bifurcation Of Two Classes Of Differential Systems With Distributed Delays

In this thesis, we mainly investigate the dynamics of two classes of differential equations with distributed delays: cellular neural networks with delays and a regulated Logistic growth model with distributed delays. The topics include global asymptotical stability and global exponential stability of equilibrium, periodic solutions, Hopf bifurcation and so on, and meanwhile the possible effects of delays on the dynamical behaviors are disscused.The whole thesis consists of five chapters.Chaper 1 is introduction, which introduces concisely the development history of differential equations with delays, especially for delayed cellular neural networks and delayed differential equations in population biology, as well as the present development of relevant research subjects and the main work done in this thesis.In the second chapter, we consider the asymptotical stability and exponential stability of neural networks with distributed delays. For asymptotical stability analysis, based on linear matrix inequality techeniques, we convert the stability problem into the feasibility of couples of linear matrix inequalities. On the other hand, we have obtained some algebraic sufficient conditions by constructing suitable Lyapunov functions. In addition, the existence and stability of peoriodic solutions is also considered for the systems with periodic input.Exponential convergence estimates for neural networks with discrete and distributed delays is presented in chapter 3. This chapter is an extension of chapter 2; we not noly consider the exponential stability and obtain some sufficient conditions, but also derive the speed of convergence rate. For estimating the convergence rate, we only need to solve an optimization problem subject to the upper bound of convergence rate under the constraits of a set of linear matrix inequality. Comparasions are made in the numerical examples and shown that the system is exponentially stable when delays are limited in a range. In addition, we also observe the Hopf bifurcation phenomenon by examing the effects of delays on neural networks without theoretical analysis, which futher expatiats on the aftereffect for system's dynamical behaviors.In chapter 4, we consider the stability and Hopf bifurcation of a regulated Logistic growth model with distributed delays. In section 4.1, the research purpose and background of the Logistic mode is presented and the linear stability is considered in section 4.2 by investigating the distribution of roots for a trancendental equation. The theoritical analysis for the direction and stability of Hopf bifurcating periodic solutions are discussed in detail in section 4.3.In the fifth chapter, we summarize the whole work of the thesis and point out the developing directions that needs to be further exporied in the future.