Cluster synchronization in community networks with nonidentical nodes

Abstract

In this paper dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community are considered. The cluster synchronization of these networks with or without time delay is studied by using some feedback control schemes. Several sufficient conditions for achieving cluster synchronization are obtained analytically and are further verified numerically by some examples with chaotic or nonchaotic nodes. In addition, an essential relation between synchronization dynamics and local dynamics is found by detailed analysis of dynamical networks without delay through the stage detection of cluster synchronization.

Group synchronization of diffusively coupled harmonic oscillators

“…. , P q }, if each of the subgraph G i with respect to P i has a directed spanning tree, and the conditions (i), (ii) of Theorem 4.3 hold, then the system (2) with the control input (3) can solve group synchronization problem asymptotically, and the synchronization state of the lth group can be explicitly given by (20).…”

“…The first strategy is to aim at nonidentical agent dynamics in different groups with positive couplings [14,19,21,25]. The other is to focus on identical agent dynamics with positive and negative couplings among the groups [12,20,22]. However, so far, there has been very little work to fully address group synchronization of coupled harmonic oscillators with directed interaction topology.…”

“…As a result, group or cluster synchronization problem for different kinds of multi-agent systems has recently been a rather significant topic in both theoretical research and practical applications. It is reported that, in general, two designed strategies have been effectively employed to realize group or cluster synchronization of networked multi-agent systems modelled by first-order or second-order integrator dynamics [12,14,20,21,22,25]. The first strategy is to aim at nonidentical agent dynamics in different groups with positive couplings [14,19,21,25].…”

On the formulation and solution of the isochronal synchronization stability problem in delay-coupled complex networks

“…22 Towards the solution of the network synchronization problem, control techniques are generally used to drive the oscillators, clusters or networks to different forms of synchrony, including isochronal. 7,8,[10][11][12][23][24][25][26][27][28] In some cases, this is done by assuming an external input reference signal in the control loop, such that zero-lag synchronization is achieved on the basis of a common target trajectory. [23][24][25][26][27] Within such framework, analytic criteria were derived so that parameters of the network are adjusted (e.g., feedback matrices), the a)…”

Pinning synchronization of the drive and response dynamical networks with lag

“…In the past few years, several kinds of network models have been proposed for the purpose of describing the real world more realistic [1,5,6,7,8]. In the complex dynamical networks, one of the most remarkable phenomena is their spontaneous synchronization, and so many types of synchronization, such as complete synchronization [9], phase synchronization [10], projective synchronization [11,12,13], impulsive synchronization [14,15,16], and cluster synchronization [17,18,19] have deeply caught the eyes of the researchers in the past few decades. However, the phenomenon of synchronization also can be classified into 'inner synchronization' [14,20] and 'outer synchronization' [21,22] from another point of view.…”