Synchronization Analysis For Several Classes Of Stochastic Complex Dynamical Networks

In recent years, complex networks have received much attention of the re-searchers from diferent felds, such as physics, mathematics, engineering，socialscience and biology etc. Especially, the synchronization for various complex net-works is a hot topic. Research on the synchronization and control of complexnetworks is not only of great theoretical value but also of practice signifcation.As real network systems are often subject to stochastic perturbations, some dif-ferent types of time delays as well as agent diferences, the analysis problemsare investigated by using the Lyapunov functional method, matrix theory, graphtheory as well as the tools from stochastic diferential equations. Finally, thestrong limit theorems for nonhomogeneous Markov chains indexed by a Cayleytree are investigated. The main contents are as follows:In the frst chapter, the basic knowledge and the development status ofcontrol and synchronization of complex networks are summarized. Some prelim-inaries, the main contents and the contribution of this thesis are also given atlast.In the second chapter, the synchronization problems of stochastic complexdynamical networks with multiple time-varying delays are investigated. By usingthe approach basing on the scramblingness property of the network’s weightedadjacency matrix, the condition ensuring the almost sure exponential synchro-nization of the network is obtained. Moreover, the result is extended to networkswith stochastically switching topology. The maximal allowable upper bounds ofcommunication delays are obtained in each case.In the third chapter, the problem of fnite-time synchronization of stochasticnonlinear multi-agent networks is discussed. By using suitable controllers withstochastic communication noises, sufcient conditions for achieving almost sure fnite-time synchronization under fve diferent kinds of communication topologiesare derived by using Lyapunov functional method and fnite-time stability theoryof stochastic diferential equations.The forth chapter investigates the bounded synchronization problem of multi-agent networks consisting of nonlinear nonidentical agent dynamics with stochas-tic noises and communication time delays. By using tools from diferential equa-tions and stochastic calculus, together with results from matrix theory and alge-braic graph theory, a sufcient condition under which the considered system canachieve mean-square bounded synchronization is derived. Moreover, the result isextended to networks with arbitrary deterministic switching topology. Explicitdelay upper bounds for guaranteeing the mean-square bounded synchronizationare obtained in each case.In the ffth chapter, the fnite-time outer synchronization between two com-plex networks with Markovian intermittent time delays is investigated. Here, thetime delay is determined by a irreducible continuous-time homogeneous Markovchain with state space {0,1}, that is, state0corresponds to the delay τ=0,and state1corresponds to the delay τ>0. By using suitable controllers, suf-cient conditions for the almost sure fnite-time outer synchronization are derivedbased on the stochastic process analysis of the state evolution of the networks.The obtained result covers the outer synchronization criterion for networks with-out communication delays as a special case.The sixth chapter investigates the strong limit theorems for nonhomogeneousMarkov chains indexed by a Cayley tree. By constructing a suitable martingale,the local convergence theorem for nonhomogeneous Markov chains indexed byCayley tree is frst studied, then some limit theorems for the frequencies of occur-rence of states and ordered couples of states for nonhomogeneous Markov chainsindexed by that tree are derived, fnally the strong law of large numbers and theShannon–McMillan theorem with a.e. convergence for nonhomogeneous Markovchains indexed by Cayley tree are obtained. In fact, our present outcomes cangeneralize the relative results of level nonhomogeneous Markov chains indexedby a Cayley tree.In the seventh chapter, the conclusions and some topics for future work aregiven.