Complex Network Synchronization Analysis

In this thesis,synchronization behaviors of some complex networks are discussed.Firstly,linearly coupled systems are studied.Before analyzing synchronization,we make some preparations:discussing continuation of solutions of a general coupled model;giving a sufficient and necessary condition of synchronization manifolds being invariant manifolds for linearly coupled systems.After these preparations,we study full synchronization and cluster synchronization of linearly coupled systems.For linearly coupled systems with time-varying coupling,by using the properties of Householder transformations,we obtain several full synchronization criteria which are based on the relationship among the elements of the coupling matrix.For linearly coupled systems with constant coupling,by means of theoretical analysis and numerical simulations,we investigate how topological structure affects cluster synchronization.Because it is difficult to simply use linear coupling to realize a desired cluster synchronization pattern independent of initial values, an alternative approach to realizing cluster synchronization is proposed.By introducing pinning controllers into linearly coupled systems,the realization problem for cluster synchronization is transformed to a pinning control problem.Sufficient conditions for ensuring the pinning control performance are given.A method of adapting the coupling strength is also provided to refine the result.Secondly,the MS model with delayed coupling is studied.Mirollo and Strogatz proved that for almost all initial conditions,the MS model with undelayed coupling eventually becomes completely synchronized.However,by numerical simulations,researchers found that the synchronization behavior of the MS model changes a lot,if transmission delays are introduced.In this thesis,rigorous mathematical analysis is presented for complete synchronization,asymptotic synchronization and weak asymptotic synchronization of the MS model with delayed coupling.Furthermore,some firing properties are obtained.