Multistability Of Complex-valued Recurrent Neural Network

Recurrent neural network which is also called artificial neural network,it is divided into the time recurrent neural network and the structure recurrent neural network.In recent decades,the scholars found that the recurrent neural network can be widely used in various fields,such as fault diagnosis,signal processing,image processing,pattern recognition,parallel computing,communications and industrial automation and so on.With the deepening of the research,the stability problems of the recurrent neural network gradually become the focus of many studies,and the scholars obtained some sufficient conditions about the stability of different types,including absolute stability,asymptotic stability,exponential stability and so on.Among them,the scholars prefer to study exponential stability of recurrent neural networks.Because for the recurrent neural network and how to converge to equilibrium point,the exponential stability can provide faster convergence speed and the attenuation of the network information.More importantly,when the exponential stability to guaranteed,regardless of any transformation,the recurrent neural network can be quickly stable.Therefore,in this article,we will focus on research for complex exponential stability of the recurrent neural network.From the perspective of system analysis,globally stable neural network is a single stable system in the sense of Lyapunov.that is,a unique equilibrium point asymptotically attracts all trajectories of the system.However,in many practical applications,the single stable neural network is computationally constrained,therefore the more stable system is crucial in dealing with the ideal neural computation.So,it is valuable to study the multistability of the system.In this paper,we use the fixed point theorem and the mean value theorem to study the multistability of complex-valued recurrent neural network in the sense of Lyapunov.Firstly,by dividing the state space,the whole state space is decomposed into some subspaces,and then every subspace is selected for independent analysis.Based on the medium value theorem and the fixed point theorem,we prove that the network under consideration has one equilibrium point in each subspace under some simpleassumptions.These equilibria are locally exponentially stable when some conditions are satisfied.