Complex Network Synchronization And Convergence

In this dissertation, we mainly discuss two aspects of problems concerning the dynamics in complex networks. One is synchronization and the other is consensus.In the part of synchronization analysis, we investigate the following problems:·Complete synchronization. We investigate the complete synchronization in complex networks of linearly coupled dynamical systems described by dif-ferential equations with non-Lipschitz or even discontinuous righthand sides; By integrating the Fillipov theory of differential equations with discontinuous righthand sides and the classical synchronization analysis as the basic tool, we give some sufficient conditions for complete synchronization of such net-works. These conditions can be seen as a generalization of previous results on synchronization of complex networks of linearly coupled dynamical systems described by differential equations with continuous righthand sides.·Cluster synchronization. We investigate cluster synchronization of lin-early coupled nonidentical dynamical systems. In such networks, the nodes of the complex networks are divided into serval clusters. The nodes in the same cluster have the same dynamical properties, while different clusters have distinguishable nodes. Under this framework, we investigate the cluster syn-chronization in both discrete-time and continuous-time network models. In both cases, we provide the sufficient and necessary conditions for the cluster synchronization of the networks. Furthermore, we propose an adaptive control scheme to realize cluster synchronization in these networks.·Synchronization under time-varying coupling We investigate complete synchronization in networks with time-varying coupling. First, we give a suf-ficient condition for a class of such networks to achieve complete synchroniza-tion. Then, we discuss two special stochastic switching processes:independent and identically distributed process and Markov process. In both cases, we have give sufficient conditions for the network to achieve complete synchronization almost surely.In the part of consensus analysis, we investigate the following consensus prob-lems in networks of multiagents:·Consensus in networks with cooperation and competition. It means that the underlying graph topologies can have both positive and negative weights, which can be used to describe cooperative and competitive trends in the networks. We give sufficient conditions for almost sure consensus in both discrete-time and continuous-time network models with general stochastically switching topologies. And the results are applied to the switching topologies of independent and identically distributed process and Markov chains.·Consensus in networks with adapted switching processes. When the weights are nonnegative, we use an adapted process to describe the topology switching process. Under this framework, we provide sufficient conditions for almost sure consensus and moment consensus, called Lp consensus, which is the first time to be introduced and is equivalent to almost sure consensus in our case. The conditions are in terms of conditional expectations of the union of the graphs with a fixed length and include both discrete-time and continuous-time network models. And the results apply to the switching topologies of independent and identically distributed process and Markov chains.