Local Complete Synchronization Analysis Of Linearly Coupled Ordinary Differential Systems With Time-varying Couplings

In this paper, we study the local complete synchronization of the complex dy-namical networks described by linearly coupled ordinary differential equation systems (LCODEs). Here, the coupling is time-varying in both the network structure and the reaction dynamics. Inspired by our previous paper [21], the extended Hajnal diameter is introduced and used to measure the synchronization in a general differential system. We find that the Hajnal diameter of the linear system induced by the time-varying cou-pling matrix and the largest Lyapunov exponent of the synchronized system play the key roles in synchronization analysis of LCODEs with identity inner coupling matrix. Firstly, we prove that a general coupled system is synchronized if its Hajnal diameter is less than 1. Secondly, we gain the sufficient condition to judge whether LCODEs can be synchronized or not and its equivalent condition. Thirdly, as an application, we obtain a general sufficient condition guaranteeing directed time-varying graph to reach consensus. Finally, an example with numerical simulation is provided to show the effectiveness the theoretical results.