Transition from the self-organized to the driven dynamical clusters

Singh, Aradhana; Jalan, Sarika

doi:10.1016/j.physa.2012.07.055

Cited by 4 publications

(5 citation statements)

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“…Cluster synchronization corresponds to a state where some of the nodes in a network are (phase) synchronized with each other, while they are not (phase) synchronized with the rest of the nodes of the network [31]. Depending on the mechanism of cluster formation, as discussed in the Introduction, there can be SO, D, or mixed clusters [6][7][8]14,15,32]. In the numerical investigation presented here, we consider sparse networks, primarily in order to avoid global synchronization (as the present paper focuses on cluster synchronization) and, more importantly, to have a better understanding of the mechanisms underlying cluster synchronization.…”

Section: Synchronized and Phase Synchronized Clustersmentioning

confidence: 99%

Impact of heterogeneous delays on cluster synchronization

Jalan

Singh

2014

Phys. Rev. E

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We investigate cluster synchronization in coupled map networks in the presence of heterogeneous delays. We find that while the parity of heterogeneous delays plays a crucial role in determining the mechanism of cluster formation, the cluster synchronizability of the network gets affected by the amount of heterogeneity. In addition, heterogeneity in delays induces a rich cluster pattern as compared to homogeneous delays. The complete bipartite network stands as an extreme example of this richness, where robust ideal driven clusters observed for the undelayed and homogeneously delayed cases dismantle, yielding versatile cluster patterns as heterogeneity in the delay is introduced. We provide arguments behind this behavior using a Lyapunov function analysis. Furthermore, the interplay between the number of connections in the network and the amount of heterogeneity plays an important role in deciding the cluster formation.

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Section: Synchronized and Phase Synchronized Clustersmentioning

confidence: 99%

Impact of heterogeneous delays on cluster synchronization

Jalan

Singh

2014

Phys. Rev. E

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“…A Lyapunov function analysis can be carried out for delayed case in a very similar fashion as for τ = 0 described in [22], and for a pair of synchronized nodes on a bipartite network can be written as:…”

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“…Hence, delay does not affect synchronization between the nodes which are not directly connected [22], and only comprehends its presence for those which are directly connected. As a consequence, depending upon ε and τ , it may either enhance or destroy the synchrony between them.…”

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confidence: 99%

“…Aforementioned can be explained further using the example of bipartite networks. A Lyapunov function analysis can be carried out for delayed case in a very similar fashion as for τ = 0 described in [22], and for a pair of synchronized nodes on a bipartite network can be written FIG. 3: (Color online) Three nodes schematic diagram illustrating impact of delay.…”

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confidence: 99%

“…For ideal D state, the synchronization between two nodes which are not directly connected is independent of the delay terms as the coupling terms cancel out, and only depends on ε. Hence, delay does not affect synchronization between the nodes which are not directly connected [22], and only comprehends its presence for those which are directly connected. As a consequence, depending upon ε and τ , it may either enhance or destroy the synchrony between them.…”

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confidence: 99%

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Role of delay in the mechanism of cluster formation

2013

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We study the role of delay in phase synchronization and phenomena responsible for cluster formation in delayed coupled maps on various networks. Using numerical simulations, we demonstrate that the presence of delay may change the mechanism of unit to unit interaction. At weak coupling values, same parity delays are associated with the same phenomenon of cluster formation and exhibit similar dynamical evolution. Intermediate coupling values yield rich delay-induced driven cluster patterns. A Lyapunov function analysis sheds light on the robustness of the driven clusters observed for delayed bipartite networks. Our results reveal that delay may lead to a completely different relation, between dynamical and structural clusters, than observed for the undelayed case. Studying the impact of network topology on dynamical processes is of fundamental importance for understanding the functioning of many real world complex networks [1]. The dynamical behavior of a system depends on the collective behavior of its individual units. One of the most fascinating emergent behavior of interacting chaotic units is the observation of synchronization [2]. In general, synchronization may lead to more complicated patterns including clusters [3][4][5]. The interplay between underlying network structure and dynamical clusters has been the prime area of focus for the last two decades [6]. Furthermore, communication delay naturally arises in extended systems [7]. A delay gives rise to many new phenomena in dynamical systems such as oscillation death, stabilizing periodic orbits, enhancement or suppression of synchronization, chimera state, etc [8][9][10][11][12][13][14].In this paper, we study the impact of delay on the phenomenon of phase synchronized clusters in coupled map networks. We investigate the formation of clusters on various networks namely, 1-d lattice, small-world, random, scale-free and bipartite networks [15], and provide a Lyapunov function analysis for bipartite networks to explain possible reasons behind the role of a delay on synchronized clusters. So far, studies on delayed coupled dynamical systems mostly concentrated on a global synchronized state, except a few recent studies which have focused on pattern formation or clustered states [4,5,16,17]. These studies have revealed that delay emulates qualitative changes in clustered state, whereas mechanism of delayed unit to unit interactions has not been investigated so far.Previous studies on undelayed coupled systems have identified two different phenomena for synchronization namely, the driven (D) and the self-organized (SO) [3]. * sarika@iiti.ac.in SO (D) synchronization refer to the state when clusters are formed because of intra-cluster (inter-cluster) couplings. Here, we report that a delay can play a crucial role in the formation of clusters as well as the phenomenon behind it. The formation of delay-induced synchronized clusters may be because of inter-cluster couplings, instead of coupling between synchronized units [5,16]. Introduction of a delay may result ...

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Cluster synchronization in directed networks of non-identical systems with noises via random pinning control

Cao

et al. 2014

Physica A: Statistical Mechanics and its Applications

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Transition from the self-organized to the driven dynamical clusters

Cited by 4 publications

References 59 publications

Impact of heterogeneous delays on cluster synchronization

Impact of heterogeneous delays on cluster synchronization

Role of delay in the mechanism of cluster formation

Cluster synchronization in directed networks of non-identical systems with noises via random pinning control

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