Study On Dynamical Behavior Of A Class Of Static Recurrent Neural Networks With Time Delays

In the recent decades, more and more scientific scholars are interested in the theory and application of artificial neural networks because of its nonlinear processing system, which is one of the most active researches in the area of the nonlinear science. Artificial neural networks can be divided into two types, one is the engineering neural networks which are used for information processing by software or hardware, the other is mathematical neural networks which is mathematical model of the former and usually called recurrent neural networks. Investigations of dynamical characteristic of the latter can provide theoretical support and guarantee for the former. Based on the difference of basic variables, the mathematical models of recurrent neural networks can be divided into two types—local field recurrent neural network models and static recurrent neural network models. Most current research about recurrent neural networks focused on the local field models, few paid attention to the static models. However, static models are widely representive. Many useful neural networks are modeled as static models. It is important to investigate the static models.In the realization of electronic neural networks, time delays are unavoidable due to the finite switching speed of amplifiers. For this reason, classes of static recurrent neural networks with time delays are investigated in this paper. Like the time delays, impulses and stochastic perturbations exist in the realization of electronic neural networks, they often exist in the same system with time delay. So in this paper, time delays, impulses and stochastic perturbations are considered in the investigated static recurrent neural networks.This paper is composed of seven chapters and main results are described as follows:In Chapter one, the historical background and the recent development of static recurrent neural networks is introduced, and the main results of this paper are also briefly introduced. In Chapter two, by using the fixed point theory, properties of matrix M, the Lyapunov function and differential inequality, firstly, the existence and global robust exponential stability of periodic solution for the static neural network with time-delays is investigated, after that, the existence and global exponential stability of almost periodic solution for the static neural network with distributed delays is investigated, and some sufficient criteria are established, which improves and generalizes some known resultsIn Chapter 3, the author investigates the global exponential stability of the equilibrium point and periodic solutions for impulsive static neural networks with time delays or distributed delays by using the fixed point theory and Lyapunov-like stability theorem, and two numerical examples are given for illustration of the theoretical results.In Chapter 4, stochastic perturbations to the stability property of static recurrent neural networks with discrete and continuously distributed delay are considered. By establishing L-operator inequality, Ito's formula and semimartingal convergence theorem, we obtain some sufficient criteria to check the almost sure exponential stability of the each system.In Chapter 5, the author investigates the system of the static recurrent neural networks with discrete delay and Markovian switching and the system of the stochastic static recurrent neural networks with distributed delay and Markovian switching, sufficient criteria to check the global exponential stability of the each systems are established by using a new Lyapunov-krasovskii functional, linear matrix inequality and the properties of Markovian chain.In Chapter 6, first, the exponential stability of the delayed static reaction-diffusion recurrent neural networks is investigated by generalized Halanay inequality, after that, due to the properties of Markovian chain, a sufficient condition to check the exponential stability of the delayed static reaction-diffusion recurrent neural networks with Markovian switching is obtained.In Chapter 7, the studying contents and the main results of this paper are briefly summarized, further more, some research prospects are also proposed.