Fixed-time synchronization of multiplex networks by sliding mode control

Abstract

Multiplex networks involve different types of synchronization due to their complex spatial structure. How to control multiplex networks to achieve different types of synchronization is an interesting topic. This paper considers the fixed-time synchronization of multiplex networks under sliding mode control (SMC). Firstly, for realizing three types of synchronization of multiplex networks in a fixed time, a unified sliding mode surface (SMS) is established. After that, based on the theory of SMC, a sliding mode controller (SMCr) which is more intelligent and has a simpler form than those in the existing literature is put forward for multiplex networks. It can not only guarantee the emergence of sliding mode motion, but also can realize three different kinds of synchronization by adjusting some parameters or even one parameter of the controller. Based on some theories of fixed-time stability, this paper deduces several sufficient conditions for the trajectories of the system to reach the preset SMS in a fixed time, and derives some sufficient conditions for multiplex networks to realize three different types of fixed-time synchronization. At the same time, the settling time which can reveal what factors determine the fixed-time synchronization in multiplex networks is obtained. Finally, in numerical simulations, different chaotic systems are set for each layer of multiplex networks to represent the nodes of different layers, which can prove that the theoretical results are practical and effective.