Synchronisation in complex networks of coupled systems with directed topologies

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. 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. 1. Introduction Complex networks have been widely used to model complex systems in science, engineering, biology, etc., since a mathematical term 'graph' is used to describe the interactions between individuals in complex systems. In such graphs, vertices represent individuals in the complex system and edges represent interactions among individuals. Typical examples of complex networks include Internet, WWW, cellular and metabolic networks, neural nets and so on (Albert and Baraba´si 2002). The complexity of such networks lies in two aspects: structure and dynamics. And these two aspects are linked to each other. Besides regular graph topologies, such as the k-nearest neighbourhood, complete graph and star-like wiring, complex structures , especially including randomness and evolution, have attracted …