Cluster Synchronization of Time‐Varying Delays Coupled Complex Networkswith Nonidentical Dynamical Nodes

Abstract

This paper investigates a new cluster synchronization scheme in the nonlinear coupled complex dynamical networks with nonidentical nodes. The controllers are designed based on the community structure of the networks; some sufficient criteria are derived to ensure cluster synchronization of the network model. Particularly, the weight configuration matrix is not assumed to be symmetric, irreducible. The numerical simulations are performed to verify the effectiveness of the theoretical results. 2 Journal of Applied Mathematics is presented. The method, which may find applications in communication engineering and other fields of science and technology, is illustrated through concrete examples of coupled biological cell models. Wu and Lu 27 investigated cluster synchronization in the adaptive complex dynamical networks with nonidentical nodes by a local control method and a novel adaptive strategy for the coupling strengths of the networks. Ma et al. 28 proposed cluster synchronization scheme via dominant intracouplings and common intercluster couplings. Sorrentino and Ott 29 studied local cluster synchronization for bipartite systems, where no intracluster couplings driving scheme exist. Chen and Lu 30 investigated global cluster synchronization in networks of two clusters with inter-and intracluster couplings. Belykh et al. 31, 26 studied this problem in 1D and 2D lattices of coupled identical dynamical systems. Lu et al. 32 studied the cluster synchronization of general networks with nonidentical clusters and derived sufficient conditions for achieving local cluster synchronization of networks. Recently, Wang et al. 33 considered the cluster synchronization of dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community by using pinning control schemes. However, there is few theoretical result on the cluster synchronization of nonlinear coupled complex networks with time-varying delays coupling and time-varying delays in nonidentical dynamical nodes.Motivated by the above discussions, this paper investigates cluster synchronization in the nonlinear coupled complex dynamical networks with nonidentical nodes. The controllers are designed based on the community structure of the networks; some sufficient criteria are derived to ensure cluster synchronization in nonlinear coupled complex dynamical networks with time-varying delays coupling and time-varying delays in dynamic nodes. Particularly the weight configuration matrix is not assumed to be symmetric, irreducible.The paper is organized as follows: the network model is introduced followed by some definitions, lemmas, and hypotheses in Section 2. The cluster synchronization of the complex coupled networks is discussed in Section 3. Simulations are obtained in Section 4. Finally, in Section 5 the various conclusions are discussed.

H∞-Based Pinning Synchronization of General Complex Dynamical Networks with Coupling Delays

“…, , then ‖̂̂( )‖ ∞ < follows. To summarize, the error dynamical system (13) is asymptotically stable and ‖ ( )‖ ∞ < , if the equations (18) are all asymptotically stable and satisfy the ∞ index .…”

“…Proof. Due to Lemma 5, all equations in the system (18) are asymptotically stable and satisfy the ∞ performance index , which implies that the error dynamical system (13) is asymptotically stable and ‖ ( )‖ ∞ < for all ∈ [0, * ].…”

“…Typical examples include the Internet, which is an enormous network of many routers connected by physical or wireless links with information packets flowing on them, and traffic and transportation network [3][4][5]. Recently, dynamical processes of the complex dynamical networks such as synchronization have been extensively investigated [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The synchronization discussed here is a kind of typical collective behaviors and basic motions in nature [22].…”

Projective Synchronization Analysis of Drive-Response Coupled Dynamical Network with Multiple Time-Varying Delays via Impulsive Control

“…Meanwhile, many kinds of synchronization have been proposed, such as complete synchronization, phase synchronization, lag synchronization, anticipated synchronization, cluster synchronization, generalized synchronization, and projective synchronization. And various control methods such as adaptive control [12][13][14][15], impulsive control [16][17][18][19][20], pinning control [21][22][23][24][25], and intermittent control [26,27] have been reported to achieve the different kinds of synchronization for complex networks.…”