Dynamics Of Transiently Chaotic Neural Networks And A Class Of Delay Differential Equations

In this thesis, we firstly introduce the models and the research progress of the discrete-time neural networks and the delay differential equations (DDE) in recent years. On the basis of them, we give a proof of the existence of an equilibrium point by Schauder fixed-point principle and a generalized sufficient condition that guarantees the asymptotical stability of it in transiently chaotic neural networks (TCNN) by using the Lyapunov function. What's more, we present a sufficient condition that guarantees the existence of chaos in the TCNN. Secondly, we define a discrete Lyapunov functional for the DDE with local positive or negative feedback in the delay term, and give a proof of the validate of this definition and some basis properties of this functional. Furthermore, we deduced some sufficient conditions that guarantees not only the existence of the global attractor and some properties of limits set, but also the Morse Decompositions of the global attractor. In particular, we study the properties of the solutions of the DDE with local monotone in the delay term. Finally, based on the previous results, we give the conditions of the existence of the global attractors and their concrete structures for various particular delay differential equations in applications.This thesis is divided into five chapters. In Chapter 1,we introduce the mathematical models and the research progress for the discrete-time neural networks and the DDE in recent years. We show that it is necessary to analysis the stability and the complex dynamics in the concrete mathematical models.In Chapter 2, we firstly introduce the mathematical models of some discrete-time neural networks and give a proof of both the existence of anequilibrium point by Schauder fixed-point principle and a generalized sufficient condition that guarantees the asymptotical stability of TCNN with asymmetric connection wrights matrix by using a new Lyapunov function. We further study the stability of an equilibrium in TCNN with the connection wrights matrix in form of interval matrix. Secondly, on the basis of studying the problem in the proof of both Marotto theorem , we derived the sufficient conditions that guarantees the existence of chaos in the sense of Li-Yorke theorem in one-dimensional TCNN, which leads to a fact that Aihara has demonstrated by numerical method. Several examples and numerical simulations are shown to illustrate and reinforce our theory.In Chapter 3, we firstly give both the definition of the discrete Lyapunov functional and the proof of some properties of it for the DDE with local positive or negative feedback in the delay term. Secondly, we introduce the definition of the global attractor. And we give some sufficient conditions that guarantees the existence of the global attractor for the DDE. By the way, we estimate the boundedness of solutions for the DDE. We also derived the Morse Decompositions of the global attractor for the DDE with local positive or negative feedback in the delay term. Finally, some properties of limits set of the solution of the DDE with local monotone in the delay term are given. Moreover, using the above discrete Lyapunov functional, we prove that the Poincare-Bendixson theorem holds for some solutions of this DDE.In Chapter 4, detail analysis of the global attractor for three particular classes of delay differential equations in concrete applications are given. Various results on the existence and the Morse Decompositions of global attractor for the DDEs in neural networks or optically bistable device are presented. We give the condition of Hopf bifurcation when the delay is a parameter in thedelay neural networks. We also give some sufficient conditions that guarantees the existence of periodic solutions and the asymptotical behavior of all positive solutions about their positive steady state in a model for population dynamics of isolated species with limited food supply and nonlinear advertising captial model with time delay feedback.At the end of this dissertation, we list some pro...