Multi-Stability And Multi-Synchronization Of Time-Delayed Memristive Neural Networks

The memristor is a new type of passive two-terminal non-volatile circuit element which integrates computing and storage.It has the advantages of small size(nanometer level),low power consumption,high ductility and easy integration,and is an ideal device for simulating biological neuron synapses.In recent years,novel artificial neural networks based on memristor have developed rapidly.Compared with traditional neural networks,memristive neural networks have faster computing speed and stronger data storage capacity,and have better performance in associative memory,neuromorphic computing,pattern recognition,brain science and other aspects.The study of the dynamic characteristics of memristive neural networks is an important way to reveal its working principle and function,and also the premise and foundation of its engineering application.In the dynamic characteristics of neural networks,the equilibrium point of neural networks corresponds to the memory of neural network under external excitation,the number of equilibrium points corresponds to the memory capacity of neural networks,and the local convergence rate of equilibrium points corresponds to the reaction speed of memory.The multi-stability of neural networks can enhance the information storage capacity of neural networks.At the same time,experimental and theoretical analysis show that the mammalian brain not only has the ability of multi-state associative memory,but also has the ability to realize the synchronization of oscillating neurons through selective perceptual attention.The synchronization of neural networks with multiple stability under coupling is called multiple synchronization.The research on the multi-stability and multi-synchronization of memristive neural networks is conducive to understanding the human brain and promoting the development and application of memristive neural networks in the fields of neuromorphic computing and brain-like science.This dissertation focused on the multi-stability and multi-synchronization problems of delayed memristive neural networks,and presented some novel criteria based on Filippov stability theory,which enriched the study for dynamics of neural networks.The main contents of this dissertation are summarized as follows:(1)The multi-stability and multi-synchronization of a class of memristive neural networks with unbounded time-varying delays are studied.Firstly,in order to expand the storage capacity of neural networks,a sufficient condition for the coexistence of multiple equilibrium states in a class of piecewise linear activation function for time-delay memristive neural networks is derived by combining the state-space segmentation method and Brouwer fixed point theory.Secondly,aiming at the problem of system instability caused by unbounded time-varying delay in neural networks,multi-stability criteria for memristive neural networks with unbounded time-varying delays are obtained under?-stability theory.According to the different upper bound of time delays,the system can achieve exponential stability,power stability,logarithmic stability and other different convergence rates.Then,based on the multi-stability results,using impulsive control and linear matrix inequality(LMI),the criterion for the multi-synchronization of memristive neural networks with unbounded time-varying delay is derived.In addition,considering the possible problems of system disturbances and parameter mismatches in practical applications,a more realistic model of time-delayed impulsive coupled memristive neural network with random disturbances and parameter mismatches is constructed,and the robust multi-synchronization conditions of the system model are given.(2)The multi-stability and multi-synchronization of a class of memristive CohenGrossberg(CG)neural networks with time-varying delays and distributed delays are studied.Firstly,in order to solve the problem of multiple stability of delayed memristive CG neural networks with arbitrary continuous activation function,a criterion for multiple stability of memristive CG neural networks with mixed delays is proposed according to the geometric structure of activation function.The restriction that the activation function is piecewise linear is eliminated,which makes the criterion of multi-stability more extended and universal.Based on the above results,the multi-synchronization problem of memristive CG neural networks with mixed time-delays under linear coupling is further studied.The corresponding synchronization controllers are designed in the case of measurable and continuously differentiable delays and unmeasurable delays respectively,and a unique Lyapunov-Krasovskii functional is constructed.Sufficient conditions for the system to achieve finite time multiple synchronization are derived by using inverse proof method.(3)The multi-stability and multi-synchronization problems of integer-order delayed memristive neural networks are extended to fractional-order ones.A class of fractional nonlinearly coupled memristive neural network with time-delays is constructed considering the fractional capacitance variation and the nonlinear coupling relationship in complex networks.The system model consists of an independent master sub-network and several slave sub-networks coupled with the master sub-network and other sub-networks.Firstly,based on fractional Halanay inequality,Caputo derivative property and fractional differential inclusion theory,the multi-stability criterion of delayed fractional-order memristive neural networks is derived.Then,based on the coupling structure of the network,the pinning multi-synchronization conditions for the fractional-order coupled memristive neural networks under pinning control are given.In addition,aiming at the problems of low bandwidth utilization and high pressure of network information transmission in practical network system,an adaptive quantization controller is designed for the slave sub-network by combining adaptive control and quantization control theory,and a unique Lyapunov functional is constructed according to the quantization parameters of the controller to realize the adaptive multi-synchronization of the system.