Synchronisation in complex networks of coupled systems with directed topologies

Abstract

The aim of this article is to provide a systematic review on the framework to analyse synchronisation in complex networks of coupled systems with a focus on the situation of directed graphs. Transverse stability of synchronisation subspace/manifold is used to describe synchronous motions, which presents the idea for synchronisation analysis. The stability of the variational systems in the transverse directions to the synchronisation manifold is studied to obtain local synchronisation. As for global synchronisation, certain structure matrices are defined to measure the distance from the collective states of the coupled systems to the synchronisation subspace, which serves as a candidate Lyapunov function. In this way, the synchronisability of a directed graph can be denoted by extending the algebraic connectivity to the underlying graph via Rayleigh–Ritz ratio. These methods and results depict how the interaction structure among individuals affects the global dynamics. Coupling delay and time-varying couplings are also considered. Furthermore, these ideas and methods can be used to investigate synchronisation of discrete-time networks of coupled maps and pinning control problem.