Some notes for synchronization in complex networks

Despite that the investigation of synchronization in complex networks has been developed more than two decades, some basic concepts remain obscure. This paper clarifies the concepts of synchronous state, synchronous orbit, solution of single node system and the relationship among them. We state that synchronous state is general solution of the single node system, while synchronous orbit is a specific solution of it. For the network whose individual system is closed with respect to addition and scalar multiplication, synchronous orbit is the specific solution with the initial value, which is the weighted mean value of the initial values of all nodes. For the network whose single node system is nonlinear, on the other hand, is different. Seeking the synchronous orbit of the network whose single node system is nonlinear is a hard work and needs further investigation. Studying these basic concepts and the relationship among them provides us a deep understanding of synchronization in complex networks and a direction of extending them to complex multi-layer networks.