Synchronization Analysis For A Class Of Complex Dynamical Networks With Non-identical Nodes

Complex dynamical networks are very common in nature and human society, and they have been widely attended in the past few decades. In general, a complex network is a large set of interconnected nodes, in which a node is a fundamental unit, which can have different meaning in different situations. As one of the most important group behavior, synchronization of complex dynamical networks has received particular attention. This paper analyzes synchronization problem for a class of complex dynamical networks with non-identical nodes; the main work of this article can be summarized into the following two parts:First, based on the common equilibrium solutions, synchronization of complex dynamical networks with non-identical nodes and asymmetric coupling is investigated. Synchronization is transformed into the problem of asymptotical stability at the original point in this part. Aimed at the feature of asymmetrical external coupling matrix of the network, we use similarity transformation to transform the external coupling matrix into the Jordan canonical form, and the corresponding coordinate transformation is adopted to partially decouple the system. Since it is impossible to decouple the network because of the non-identical nodes, common dynamics are employed to replace the different dynamics, and the difference among the different isolated-nodes dynamics is restricted. On this basis we investigate local and global synchronization criteria for the network. The sufficient conditions derived from this part of the article, can be seen as a master stability function for asymmetric coupled complex dynamical networks with non-identical nodes, and can be converted into optimization problems. Numerical examples are given to show the effectiveness of the conclusions.Second, for the asymmetric complex dynamical networks with non-identical nodes where common equilibrium solution doesn’t exist, the average state of all nodes is selected as the target of synchronization and the error dynamical equations are established. Since the error state is unable to arrive at the original point in the transformed coordinate, according to the Lyapunov stability theory and using the ultimate boundedness theory, the global synchronization criterion in the sense of boundness of error vector’s norm is proposed. It is pointed out that the complete synchronization of the networks with the same dynamic of all the isolated nodes is a special case of the conclusion. Further, while the networks satisfy the necessary condition of the complete synchronization, it can be proved that the conditions given above could result in the complete synchronization of asymmetric complex dynamical networks with non-identical nodes. The numerical simulation results verify the validity of the conclusions.Finally, the main work of the full paper is summarized and the future researching work is pointed out.