The existence of periodic solutions for coupled systems on networks with time delays

• Graph theory, Lyapunov method and coincidence degree theory are combined together to study existence of periodic solutions of coupled systems on networks.
• A systematic approach for the existence of periodic solutions to general system is obtained.
• Sufficient conditions for the existence of periodic solutions to nonlinear coupled oscillators on a network with time delays are derived.

This paper is concerned with the existence of periodic solutions for coupled systems on networks with time delays (CSND). By the combined method of graph theory, coincidence degree theory and Lyapunov method, a systematic approach for the existence of periodic solutions to CSND is developed. We apply this approach to a coupled system of nonlinear oscillators with time delays and obtain the existence of periodic solutions. Finally, a numerical example is provided to illustrate the effectiveness of the results developed.