Counterpart synchronization of duplex networks with delayed nodes and noise perturbation

TL;DR

This paper investigates the synchronization of duplex networks with delayed nodes and noise, proposing a control method based on LaSalle's invariance principle to achieve counterpart synchronization across diverse network configurations.

Contribution

It introduces a novel control technique for counterpart synchronization in duplex networks with delays and noise, applicable to various node dynamics and network topologies.

Findings

Proposed a sufficient condition for synchronization.

Derived two corollaries for special cases.

Numerical examples confirm effectiveness.

Abstract

In the real world, many complex systems are represented not by single networks but rather by sets of interdependent ones. In these specific networks, nodes in one network mutually interact with nodes in other networks. This paper focuses on a simple representative case of two-layer networks (the so-called duplex networks) with unidirectional inter-layer couplings. That is, each node in one network depends on a counterpart in the other network. Accordingly, the former network is called the response layer and the latter network is the drive layer. Specifically, synchronization between each node in the drive layer and its counterpart in the response layer (counterpart synchronization, or CS) of this sort of duplex networks with delayed nodes and noise perturbation is investigated. Based on the LaSalle-type invariance principle, a control technique is proposed and a sufficient condition is developed for realizing counterpart synchronization of duplex networks. Furthermore, two corollaries are derived as special cases. In addition, node dynamics within each layer can be various and topologies of the two layers are not necessarily identical. Therefore, the proposed synchronization method can be applied to a wide range of multiplex networks. Numerical examples are provided to illustrate the feasibility and effectiveness of the results.