Synchronization Analysis For Stochastic Multilayer Complex Network

Research on complex network has become a hot topic,because of its wide appli-cations on computer science,bioinformation,security on communications and social science.In the study of complex network,the synchronization problem has very important theoretical and applied significance.Besides,a single network may not be able to describe a real complex system because of the rapid development on the research of complex network.Therefore,the related research should be built on multilayer complex network.In this dissertation,we study the synchronization and control problems of several stochastic and delayed dynamical systems on some multilayer networks.Two types of multilayer networks are considered.One is the multipex network(Chapter 2 and Chapter 3),which contains several subnetworks or layers representing different functions or topologies,and the states of the nodes on different layers may be the same or different.The other one(Chapter 4 and Chapter 5)is the duplex network which contains two layers or subnetworks,in which the states of each node on different layers are different.The controllers are,state feedback controller,adaptive state feed back controller,pinning controller and impulsive controller.The tools to study the synchronization come from dynamical theory,matrix theory,graph theory and simulation,e.g.,Lyapunov stability theory,LaSalle invariance principle,linear matrix inequalities,impulsive Halanay inequality and LMI toolbox of Matlab.The types of synchronization are,inner synchronization,outer synchronization,intra-layer synchronization,almost sure synchronization and exponential synchronization.Besides,for each main result,we design numerical example,solve and simulate using Matlab,and draw figures to show the feasibility and effectiveness of the result.The main innovations include the follows.All networks are multilayer,i.e.,both in Chapter 2 and Chapter 3 are multiplex networks,both in Chapter 4 and Chapter 5 are duplex networks including two layers of subnetwork.All models incorporate stochastic terms to represent the environmental noises.All the delays are time-varying.Almost all the sufficient conditions to achieve synchronization are represented by linear matrix inequalities(LMI).Otherwise if they are not linear,then we design algorithms to solve these inequalities using LMI toolbox.Specifically,the main results and innovations are summarized and listed as below.First chapter is the introduction to this dissertation.Firstly,we focus on the literature review,giving the background on the study of synchronization problem and control problem for stochastic delayed dynamical systems on multilayer complex network.Secondly,we introduce the progress on this topic,different types of synchronization phenomena,and some widely applied controllers.Thirdly,we give the preliminary including notation and lemmas from complex network,matrix theory and stochastic differential equations.Finally,we conclude this chapter with the arrangement of this dissertation.Chapter 2 is concerned with the inner synchronization and control problem of stochastic delayed multilayer networks with additive couplings.Multilayer networks are a kind of complex networks with different layers,which consist of different kinds of interactions or multiple subnetworks.Additive couplings are designed to capture the different layered connections.Based on additive couplings,several sufficient conditions are obtained to guarantee the synchronization of chaotic stochastic delayed coupled multilayer network.In essence,this type of synchronization is inner synchronization,in which the states of all nodes approach a same synchronous state.More specifically,on one hand,we obtain some sufficient conditions to guarantee that the stochastic multilayer network can be synchronized almost surely without control input.On the other hand,we propose three synchronization schemes by designing controllers.Scheme(?):It is assumed that only a part of the nodes are allowed to be controlled directly.Scheme(?):Control all nodes of the complex system by using only one controller.Scheme(?):Pinning adaptive controller.Chapter 3 concerns the outer synchronization and control problem of two stochastic dynamical systems on multilayer networks with additive couplings and time-varying delays.Specifically,it is the synchronization between states of the nodes in the two multilayer networks:a driving system and a response system In this chapter,two pinning controllers are designed to guarantee the synchronization of the stochastic multilayer network.One is the state-feedback pinning controller with constant control gains.The other one is the adaptive pinning controller with adaptive control gains.Moreover,as we prove,only selected part of the nodes to be controlled are enough to guarantee that the drive system and response network can be stochastically synchronized.Chapter 4 concerns the intra-synchronization and impulsive control problem of a stochastic delayed dynamical system on a duplex network,which is a multilayer network containing two layers of subnetworks.Different from some previous works,environmental noise is introduced into the dynamical system on the duplex network.We incorporate both the inter-layer delay and the intra-layer delay into the dynamical system.Both of the delays are time-varying.Due to the superior efficiency of impulsive control,we use it to control the intra-synchronization of the duplex network even if the dynamical system exhibits chaos phenomena.Chapter 5 concerns the intra-synchronization and continuous control problem of a stochastic delayed dynamical system on a duplex network,which is a multilayer network containing two layers of subnetworks.Different from some previous works,environmental noise is introduced into the dynamical system on the duplex network.We incorporate both the inter-layer delay and the intra-layer delay into the dynamical system.Both of the delays are time-varying.When the system does not achieve automatic intra-layer synchronization,we introduce two controllers:one is the state-feedback controller,the other is the adaptive state-feedback controller.Interestingly,we find that the intra-layer synchronization would be achieved automatically·if the inter-layer coupling strength is large enough when the time-varying inter-layer delays are absent.In the final chapter,we conclude the dissertation with the possible future directions and the prospects.