Graph Topology and Synchronization of Network Coupled Dynamic Systems with Time-delay

This paper studies the synchronization of network coupled systems consisting of many identical dynamic subsystems as well as network coupling with interaction time-delay. Based on graph theory and Lyapunov stability theory, the paper gives conditions for the total system synchronization and the boundedness of trajectories with respect to the graph structures of network coupling interaction. According to our study, it is revealed that, different from the systems without time-delay, (1) there will be a case where synchronization will occurs if the interaction strength and the number of the graph edges lie between some upper and lower bounds, (2) in regular graphs, such that complete-, star- and cycle graph, the synchronization conditon will be violated if total number of subsystems increases, (3) for the complex networks such as Small World Network and Scale Free Network, the synchronization conditon is more easier to be satisfied than regular graphs when the total number of subsystems increases.These are verified using numerical simulations.