Interlayer synchronization of duplex time-delay network with delayed pinning impulses

Abstract

This paper mainly focuses on interlayer synchronization in a duplex network setting with the intralayer coupling delay and the interlayer transmission delay via pinning impulsive control. Firstly, based on Lyapunov stability theory, a general pinning impulsive strategy guaranteeing interlayer synchronization is obtained. Then, the minimum number of pinned nodes and the maximum impulse interval needed theoretically for interlayer synchronization are obtained. Moreover, when some parameters, including the impulse strengths, the ratio of pinned nodes and the impulse interval, are fixed certain values, their influences on the stable region are further discussed. It is found that the stable region becomes smaller as the ratio of pinned nodes decreases, and gets larger as the impulse strength with no delay is closer to −1 and that with delay is closer to 0. At the same time, the small intralayer coupling delay and interlayer transmission delay can enlarge the stable region. Finally, numerical simulations are provided to verify the effectiveness and correctness of our results.