Eur. Phys. J. B 77, 257-264 (2010)
https://doi.org/10.1140/epjb/e2010-00202-7

Cluster synchronization in networks of distinct groups of maps^*

¹ School of Mathematical Sciences, Centre for Computational Systems Biology, Fudan University, Shanghai, P.R. China
² Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, P.R. China

Corresponding author: ^a wenlian.lu@gmail.com

Received: 12 December 2009
Revised: 9 April 2010
Published online: 29 June 2010

Abstract

In this paper, we study cluster synchronization in general bi-directed networks of nonidentical clusters, where all nodes in the same cluster share an identical map. Based on the transverse stability analysis, we present sufficient conditions for local cluster synchronization of networks. The conditions are composed of two factors: the common inter-cluster coupling, which ensures the existence of an invariant cluster synchronization manifold, and communication between each pair of nodes in the same cluster, which is necessary for chaos synchronization. Consequently, we propose a quantity to measure the cluster synchronizability for a network with respect to the given clusters via a function of the eigenvalues of the Laplacian corresponding to the generalized eigenspace transverse to the cluster synchronization manifold. Then, we discuss the clustering synchronous dynamics and cluster synchronizability for four artificial network models: (i) p-nearest-neighborhood graph; (ii) random clustering graph; (iii) bipartite random graph; (iv) degree-preferred growing clustering network. From these network models, we are to reveal how the intra-cluster and inter-cluster links affect the cluster synchronizability. By numerical examples, we find that for the first model, the cluster synchronizability regularly enhances with the increase of p, yet for the other three models, when the ratio of intra-cluster links and the inter-cluster links reaches certain quantity, the clustering synchronizability reaches maximal.