Study On Several Synchronization Phenomena For Networks With Chaotic Dynamics

This thesis is devoted to the study of several synchronization phenomena for some network models with chaotic dynamics. Firstly, a brief description of historical background and research progress for chaos, complex networks and synchronization is given. Secondly, generalized synchronization for a simple network consisting two dicrete systems is studied. Then, cluster synchronization for complex networks consisted of some coupling oscillators is discussed. The research focuses mainly on four parts:(1) Generalized synchronization of two discrete systems.By constructing appropriate nonlinear coupling terms, some sufficient conditions for determining the generalized synchronization between the drive and response systems are derived. In a positive invariant and bounded set, many chaotic maps satisfy the sufficient conditions. The effectiveness of the sufficient conditions is illustrated by four examples.(2) A new method to realize cluster synchronization in connected chaotic networks.By constructing a diffusively coupled matrix with cooperative and competitive weight-couplings, which represents topological structure of a network, a sufficient condition about the existence and global asymptotic stability of an arbitrarily selected cluster synchronization invariable-manifold for a connected chaotic network is derived. When the number of the clusters is greater than one, there are some invariable submanifolds in the cluster synchronization invariable-manifold. However, all of them are unstable. That is to say, arbitrarily selected cluster synchronization patterns in a connected chaotic network with diffusive coupings can be achieved via appropriate coupled schemes. When the network realizes cluster synchronization, from the characteristic of diffusive couplings, the dynamics of every node is chaotic. The correctness and effectiveness of the method is illustrated by an example.(3) Cluster synchronization for nondiffusively coupling networks. By constructing a nondiffusively coupled matrix, the existence of an arbitrarily selected cluster synchronization invariable-manifold in a network with chaotic dynamics is given. By use of matrix theory and Lyapunov function approach, a sufficient condition about the global asymptotic stability of the invariant-manifold is derived. Accordingly, it is showed that nondiffusively coupling networks can also realize cluster synchronization. However, different from the case of diffusively coupling networks, some invariable submanifolds in the cluster synchronization invariable-manifold may be global asymptotic stable. That is to say, under some conditions, two different clusters may incorporate a bigger cluster in which all nodes are complete synchronization. The correctness and effectiveness of the theoretic results is illustrated by two examples. In addition, the examples also show that, when a nondiffusively coupling network realizes cluster synchronization, some nodes can emerge the dynamics behaviors different from uncoupled dynamics behavior at individual node.(4) Cluster synchronization of a starlike complex network.By use of matrix theory and Lyapunov function approach, a sufficient condition about the existence and global asymptotic stability of a cluster synchronization invariant-manifold in a starlike complex network is derived. The effectiveness of the sufficient condition is illustrated by some examples. In addition, the examples also show that, when the starlike complex network realizes cluster synchronization, synchronization states of some nodes vary between chaotic states and nonchaotic states as the coupling strength varies, and some nodes can emerge new dynamics behaviors.