Dynamical Behavior Analysis Of Several Class Of Discrete-time Neural Network Models

In this thesis, we firstly introduce several class of discrete-time neural network models and the research progress of the neural networks. By Schauder fixed-point principle we prove the existence of an equilibrium (i.e. a fixed point) of a discrete-time neural network with generalized input-output function and by using the converse theorem of Lyapunov function we study the uniformly asymptotical stability of equilibrium in this discrete-time neural network with variable weight and give some sufficient conditions that guarantee the stability of it. Secondly, the existence of chaos in a special class of discrete-time neural network models with sinusoidal activation function in the sense of Devaney with some parameters of the systems entering some regions are rigorously presented by means of anti-integrable limit method. What's more, period-doubling bifurcation and saddle-node bifurcation in the neuron model are studied. Finally, the existence and global exponential stability of periodic solutions in a discrete-time Cohen-Grossberg neural network with variable and invariable delay are investigated by using the continuation theorem of coincidence degree theory. And sufficient conditions are given to guarantee the existence ofω—periodic solution and all other solutions are convergent to it globally exponentially.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. We illustrate several class of important discrete-time neural networks and 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 model of a discrete-time neural network with generalized input-output function. The model generalizes the input-output function in transiently chaotic neural network to a class of continuous, differentiable and monotone increasing functions. Secondly we study the uniformly asymptotical stability of equilibrium in the non-autonomous model. Finally, several examples and numerical simulations are given to illustrate and reinforce our theories.In chapter 3, we firstly introduce a specific class of discrete-time neural network models with sinusoidal activation function. This class of models have the ability of embedded pattern retrieval in the neural network beyond the conventional one with a monotonous activation function and possess a remarkably larger memory capacity than the conventional association system. Secondly, We obtain some sufficient conditions to ensure that there exists period-doubling bifurcation and saddle-node bifurcation in the neuron model. Finally, By means of anti-integrable limit method, the existence of chaos in the discrete-time neural network models in the sense of Devaney with some parameters of the systems entering some regions are rigorously presented. In addition, several concrete examples with their numerical simulations are further provided to reinforce our theoretical results.In chapter 4, we first introduce the continuation theorem of coincidence degree theory. It is very important theoretical basis for some differential equations and difference equations to prove the existence of the periodic solution. Secondly, the existence and global exponential stability of periodic solutions in a discrete-time Cohen-Grossberg neural network with variable and invariable delay are investigated by using the continuation theorem of coincidence degree theory. And sufficient conditions are given to guarantee the existence ofω—periodic solution and all other solutions are convergent to it globally exponentially. At the end of this dissertation, we list some problems for our future works which include stability and chaos in some more complex neural networks.