The existence and exponential stability of periodic solution for coupled systems on networks without strong connectedness

This paper focuses on the problem of the existence and global exponential stability of periodic solution for coupled systems with delays on networks without strong connectedness (NWSC), which extends previous results of strongly connected networks. An innovative hierarchical method is proposed to characterize a large NWSC. Then each layer consists of several independent strongly connected subnets. By using the existing results of strongly connected networks and constructing auxiliary systems, we investigate the existence of periodic solution for original coupled systems on NWSC layer by layer. Moreover, the uniqueness and global exponential stability of periodic solution are considered as well. Then the theoretical results are applied to coupled oscillators on NWSC. Finally, a numerical example is also provided to illustrate the effectiveness of the theoretical results.