Cluster synchronization in networks of coupled nonidentical dynamical systems

Abstract

In this paper, we study cluster synchronization in networks of coupled non-identical dynamical systems.The vertices in the same cluster have the same dynamics of uncoupled node system but the uncoupled node systems in different clusters are different. We present conditions guaranteeing cluster synchronization and investigate the relation between cluster synchronization and the unweighted graph topology. We indicate that two condition play key roles for cluster synchronization: the common inter-cluster coupling condition and the intra-cluster communication. From the latter one, we interpret the two well-known cluster synchronization schemes: self-organization and driving, by whether the edges of communication paths lie at interor intra-cluster. By this way, we classify clusters according to whether the set of edges inter-or intra-cluster edges are removable if wanting to keep the communication between pairs of vertices in the same cluster.Also, we propose adaptive feedback algorithms on the weights of the underlying graph, which can synchronize any bi-directed networks satisfying the two conditions above. We also give several numerical examples to illustrate the theoretical results.

Observability and coarse graining of consensus dynamics through the external equitable partition

“…where m * r = |π * r | is the size of the mEEP with respect to node r. Equation (37) states that the rank of the observability of a node under consensus dynamics is equal to the rank of the observability of its cell in the consensus dynamics of the corresponding quotient graph, and the size of the mEEP provides an upper bound for the dimensionality of the r-observable subspace.…”

“…Finding a nontrivial EEP is thus equivalent to finding a graph (i.e., the quotient graph) for which its associated consensus dynamics describes the dynamics on the lower-dimensional synchronization manifold, which is invariant under the graph Laplacian [37,38].…”

“…Hence, in this case, "cell synchronization" is maintained at all times, and the dynamics of each cell corresponds to that of the corresponding node in the quotient graph. In other words, x lies in the "cluster synchronization manifold" [37]:…”

Group synchronization of diffusively coupled harmonic oscillators

“…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].…”

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.…”