Stabilization And Synchronization Of Two Classes Of Fractional-order Memristor-based Neural Networks

The memristor is the fourth circuit element besides resistor,capacitor and inductor,and it can memorize the charge passing through it over the past time,therefore it can exhibit the nonvolatile memory characteristics.Compared with traditional resistors,memristors are able to emulate synapses in human brains more accurately.Fractional calculus can be understood as a generalization of classical integration and differentiation to the order of rational numbers.In comparison with integer-order derivative,fractional derivative possesses nonlocal property,which is characterized by infinite memory.Hence,fractional derivative has the great advantage of describing characteristics of memory and inheritance of some materials and processes.Fractional-order memristor-based neural networks(FMNNs)can be constructed by describing dynamics of artificial neural networks(ANNs)with fractional derivatives and taking memristors as connection synapses between neurons.Stabilization of FMNNs under stabilizing controllers can be utilized to solve optimization problems and two synchronous FMNNs achieved by synchronization controllers can be applied in secure communication.This thesis contains the following three parts.(1)An incommensurate fractional-order(and an integer-order)memristor-based system is directly constructed from the Chua's circuit by replacing the Chua's diode with a flux-controlled memristor characterized by a smooth quadratic nonlinearity and a negative conductance.The dynamical analysis is carried out by using the fractional-order(and integer-order)system stability theory and numerical methods such as Lyapunov exponents,bifurcation diagrams,Poincare mappings and phase portraits.Dynamical phenomena such as(inverse)period doubling bifurcation,transient chaos,and state transfer phenomenon are determined.(2)Two types of control laws(delayed state feedback control and coupling state feedback control)are designed to globally stabilize threshold FMNNs with time delay.Accordingly,two kinds of stabilization criteria(algebraic form and LMI form)are established.There are two groups of adjustable parameters in the delayed state feedback control,and one can select them flexibly to achieve the desired global asymptotic stabilization or global Mittag-Leffler stabilization in practice.Based on the coupling state feedback control,a LMIs stabilization criterion is developed for the first time with the help of the newly-established fractional-order differential inequality.(3)A slope switching FMNNs model with multiple time-varying delays is proposed and globally asymptotical synchronization is investigated.The state feedback controller is designed and a synchronization criterion in LMIs form is derived based on the estimates of interval matrix norms.The adaptive state feedback controller without time delay is designed and globally asymptotical stability of the synchronization error system is proved.