Multistability Analysis For Fractional-order Memristive Neural Networks

In recent years,some valuable results have been obtained about multistability of neural networks.However,it should be pointed out that most existing multistability results are derived under the assumption that the considered neural networks models are classical integer-order models.With the properties of memory and hereditary,fractional differential equations can better characterize complex dynamic behaviors than integer-order ones.Memristor is an abbreviation for memory and resistor,and is an electronic element that can closely simulate the brain synapses.Based on the theory of fractional calculus,the contraction mapping theorem,the fixed point theorem,the comparison principle of fractional-order linear system and Lyapunov function method,multistability of fractional-order memristive competitive neural networks and fractional-order memristive Hopfield neural networks with delays are studied,respectively.This dissertation is divided into the following four chapters:The first chapter describes the significance of memristive neural networks,the multistability of integer-order neural networks,the research significance and current research situation analysis for the stability of fractional-order memristive neural networks.Then,based on the above analysis,the main research contents and main contributions of this dissertation are also expounded.The second chapter discusses the coexistence and the complex dynamic behaviors of multiple equilibrium points in fractional-order memristive competitive neural networks with piecewise linear activation functions.Based on the characteristic of switching threshold,the contraction mapping theorem,Lyapunov function method,the theory of fractional calculus,some sufficient conditions are established to ensure that the n-dimensional fractional-order memristive competitive neural networks have 4nequilibrium points,and among them 3nequilibrium points are Mittag-Leffler stable.Compared with the conventional neural networks,the number of total equilibrium points increases to 4n from 3n,and the number of locally stable equilibrium points increases to 3nfrom 2n.Finally,the validity of the theory is verified by two examples with their computer simulations.The third chapter studies the multistability of the fractional-order memristive Hopfield neural networks with Mexican-hat-type activation function and time delays.Based on the geometrical properties of the Mexican-hat-type activation function,the fixed point theorem,Lyapunov function method,the comparison principle of fractional-order linear system and the theory of fractional calculus,it is obtained that the n-dimensional fractional-order memristive Hopfield neural networks with time delays have 5nequilibrium points,and among them 3nequilibrium points are locally stable.Finally,the validity of the theory is verified by numerical examples.The fourth chapter summarizes the research work of this dissertation and the prospect for the future work is made.