Stability And Synchronization Of Several Fractional-Order Neural Networks

Fractional calculus operator,as the real domain extension of integral calculus operator on the operation order,has the characteristics of “nonlocality”and “infinite memory”,which provides a more accurate mathematical tool for characterizing biological neuron system,intelligent control system and other practical systems as well as natural laws.Therefore,the fractional-order neural network model can more accurately describe the biological brain behavior and improve the performance of the artificial neural network algorithm.Concurrently,in the electronic realization of neural network,time-delay is inevitable,and the existence of time-delay inevitably affects the dynamic behavior of neural network.Therefore,the qualitative research of the delayed fractional-order neural network has important theoretical significance and application value.This paper focuses on the stability of three kinds of delayed fractional-order neural networks,and discusses the delayed memristor-based neural network and Hopfield neural network in the sense of Caputo fractional-order derivative,and stability and synchronization of fuzzy neural networks with time delay in the sense of Riemann-Liouville fractional-order derivative.Specifically,the main contributions of this paper include the following three aspects:(1)The global stability of the delayed fractional-order memristor-based neural networks via feedback effects is analyzed.Based on the fractional calculus theory,set-valued mapping theory,differential inclusion,comparison principle of fractional-order differential systems,Lyapunov direct method and some differential inequalities,the sufficient conditions for the existence and uniqueness of the equilibrium point of the network model are obtained,and for two kinds of output feedback inputs,the sufficient criterion of network model stability is established.(2)The synchronization of projection coefficients among states for the master-slave network of the delayed fractional-order Hopfield neural network with parameter mismatch via sliding mode effects is analyzed.By introducing the sliding mode control strategy,a class of delayed sliding mode integral switching surface is constructed.Based on the selected appropriate approach law,the complete form of the delayed sliding mode controller is given.Based on Lyapunov direct method and sliding mode technology,the dynamics of the error system's trajectory approaching to the sliding mode surface is analyzed.By using Razumikhin functional method,the stability criterion of the sliding mode motion for the solution of the error system on the sliding mode integral surface with time delay is derived,and then the sufficient conditions for projective synchronization of nonidentical network models are obtained.(3)The problem of the state for the fuzzy neural network with delay tends to be stable via impulse effects in the sense of Riemann-Liouville fractional differential operator.Firstly,a class of hybrid Lyapunov functional is constructed by using the characteristics of Riemann-Liouville derivative.Then,by using the extended fractional-order Barbarat's lemma and the qualitative theory of impulsive system,the criteria for the stability of delayed fuzzy neural network are obtained.The results show that the fast convergence of the solution for the fractional-order neural network model can be achieved under the fixedtime impulse and fuzzy logic rules.