Periodic solutions of coupled systems on networks with both time-delay and linear coupling

This paper focuses on the existence of periodic solutions for coupled systems on networks with time-delay and linear coupling (CSNDLC). Based on graph theory, coincidence degree theory and the Lyapunov method, a systematic approach for the existence of periodic solutions to CSNDLC is developed. And this approach is applied to the linear coupled oscillators with time-delay on a network. In addition, global asymptotic stability criterion for its periodic solution is obtained. Finally, a numerical example is provided to illustrate the effectiveness of the results developed.