Finite-time And Fixed-time Synchronization For Several Classes Of Complex Dynamical Systems

Chaotic synchronization is a basic behaviour of dynamical system and has become one of the main research topics.The investigation of synchronization involves many disciplines,such as: Lyapunove stability,Matrix theory,Computer science,System science,Information science and so on.At the same time,the results of synchronization have been applied in many fields including secure communications,signal processing,image encryption,pattern recognition,and biological systems,etc.It is well known that there are many typical complex dynamical systems,including neural networks and complex networks with nonlinear functions,time delays,discontinuous activation functions or stochastic noise perturbations,etc.The synchronization and control of those complex dynamical systems have attracted much attention from researchers.This paper focus on synchronization and control for several classes of neural networks and complex networks in five aspects: finite-time synchronization of memoristive neural networks with mixed delays,finite-time synchronization of discontinuous neural networks with delays and mismatched parameters,fixed-time synchronization of complex networks with stochastic noise perturbations,fixed-time synchronization of complex networks via quantized pinning control and fixed-time cluster synchronization of stochastic complex networks.The main contributions and innovations can be summarized as follows:1.We consider finite-time synchronization for memristive neural networks with mixed delays by means of systems with interval parameters.Those systems are established via the concept of Filippov solution.As a result,finite-time synchronization of the considered memristive neural networks is equivalent to the finite-time stabilization problem of the error systems with interval parameters.In addition,new quantized controllers with or without sign function are designed.By constructing Lyapunov function,developing 1-norm-based analytical methods and applying two kinds of quantized control schemes,several conditions are derived to guarantee that the memristive neural networks can be synchronized within a finite time.Moreover,a numerical example and its simulations are provided to illustrate the theoretical analysis.2.We investigate finite-time synchronization for a class of neural networks with discontinuous activation functions,time-varying discrete and unbounded distributed delays,and mismatched parameters.In order to deal with the difficulties induced by discontinuous activations,time delays,as well as mismatched parameters simultaneously,new 1-norm-based analytical techniques are developed.Both state feedback and adaptive controllers with and without sign function are designed.Based on the designed control schemes,differential inclusion theory and Lyapunov functional method,several sufficient conditions on the finite-time synchronization are obtained.The obtained results show that the controllers with sign function can reduce the conservativeness of control gains and the controllers without sign function can overcome the chattering phenomenon.Numerical examples are given to show the effectiveness of the synchronization criteria.3.We are concerned with the fixed-time synchronization of complex networks with stochastic noise perturbations.Both state feedback controllers with and without sign function are designed to realize fixed-time synchronization.The controllers without sign function can overcome the chattering effect induced by sign function,while the control parameters are general in those controllers with sign function.By constructing suitable Lyapunov function and applying the properties of Weiner process,fixed-time synchronization criteria are derived via the designed controllers with or without sign function.At the same time,the settling time are estimated.Moreover,some numerical simulations are presented to exhibit the validity of the established results.4.We study the fixed-time synchronization of complex networks via quantized pinning controllers.Both quantized control schemes with and without sign function are designed.From the analyses of this chapter,it can be seen that the control parameters are general in quantized pinning controllers with sign function,but quantized pinning controllers without sign function can be utilized to overcome the chattering phenomenon in some existing results.Based on designed Lyapunov function and different control schemes,several fixed-time synchronization criteria expressed by linear matrix inequalities are presented and the estimations of the settling time are given.In addition,numerical examples are presented to illustrate the theoretical results.5.We focus on cluster synchronization of complex networks with stochastic perturbations via fixed-time control technique.Quantized controller is designed to realize the fixed-time synchronization and improve the accuracy of settling time.Based on the Lyapunov functional and comparison system methods,several sufficient conditions on fixedtime cluster synchronization are obtained.Moreover,some fixed-time synchronization criteria are established which are special cases of main results.Finally,numerical simulations are offered to substantiate the theoretical results.