Stability Analysis Of Hopfield Neural Network

The human brain has many advanced intelligence behaviors,such as thinking,cognition,learning and memory,and research has shown that those intelligent behaviors are the result of the neural network consisting of a large number of neurons.Therefore,studying the dynamic behavior of the brain neural network is crucial to understanding how the brain generates intelligence behaviors.Artificial neural network is proposed by imitating biological neural network.The recurrent neural network is an important part of artificial neural network,and it is widely used in various fields such as pattern recognition,image processing,computer science,automatic control,etc..The dynamic analysis of artificial neural networks is an important theoretical basis for its application.In general,the network is divided into a monostability neural network and a multi-stability neural network according to the number of network's equilibria.Although the stability of neural networks has achieved great research results,there are still many problems to be solved in theoretical research.In this paper,a series of researches on the mono-stability and multi-stability of Hopfield neural network are carried out.The main research results are as follows:Firstly,the stability of one-dimensional Hopfield neural network is analyzed.We discussed the number,the approximate location,and the stability of the equilibria of one dimension of the Hopfield neural networks.Geometry method is used to discuss the number and position of the equilibria.By using Taylor expansion,we get the conditions for stability of the equilibrium point.Secondly,this paper studies the monostability of an n-dimensional Hopfield neural network.We discussed the existence,the uniqueness and the global stability of the equilibrium point of n-dimensional Hopfield neural network.The Brouwer fixed point theorem is used in the study of the existence of equilibrium point.The uniqueness of the equilibrium point is proved by algebraic inequality.The global asymptotic stability and the global exponential stability of the equilibrium point are investigated by constructing a proper Lyapunov function.Finally,this paper studies the multistability of n-dimensional Hopfield neural network.Inspired by the studies of one-dimensional Hopfield neural network,by constructing a map and combining Brouwer's fixed point theorem,we get a sufficient condition for the n-dimensional Hopfield neural network possessing 3n equilibria.And we have proved that there exist 2n asymptotically stable equilibria among the 3n equilibria by using the Gael disk theorem and the Hartman-Grobman linearization theorem,there exist 2n exponential stable equilibria among the 3n equilibria by constructing the Lyapunov function.