Impact of a leader on cluster synchronization

Abstract

We study the mechanisms of frequency synchronized cluster formation in coupled non-identical oscillators and investigate the impact of presence of a leader on the cluster synchronization. We find that the introduction of a leader, node having large parameter mismatch, induces a profound change in the cluster pattern as well as in the mechanism of the cluster formation. The emergence of a leader generates a transition from the driven to the mixed cluster state. The frequency mismatch turns out to be responsible for this transition. Additionally, for a chaotic evolution, the driven mechanism stands as a primary mechanism for the cluster formation, whereas for a periodic evolution the self-organization mechanism becomes equally responsible. The interaction between individual units of a system leads to many emerging behaviors, among which synchronization is one of the most fascinating phenomenon which has been gaining tremendous attention since the first experimental demonstration of the phenomena by Christian Huygens [1]. The unexpected sway and twist of the Millennium bridge, synchronization of the neurons in the brain and synchronous fire flies [2][3][4] are few examples of the synchronization in the real world systems. Synchronization is defined as an emergence of some relation between the functional of two processes due to interaction [3]. Earlier studies on coupled non-identical oscillators have shown that the exact synchronization is hard to achieve when there is a parameter mismatch in the local dynamics [3], rather they exhibit phase or generalized synchronization [5,6]. Additionally, there exists a nontrivial transition to the global phase synchronization in a population of globally coupled chaotic non-identical oscillators [7]. Moreover, cluster pattern and frequency of the nodes in a cluster have been shown to be controlled by local external forcing as well as by changing the network architecture [8]. Furthermore, the coupled oscillators with heterogeneous coupling have been reported to exhibit synchronization triggered by the oscillators having strong couplings, further facilitating the synchronization among the nodes having weak coupling [9].We present results of cluster synchronization due to its importance and occurrence in various real world systems represented in terms of interacting units [10]. We study mechanisms of formation of frequency synchronized clusters in coupled non-identical oscillators and investigate the influence of a leader on the dynamical evolution of other nodes. One possible way of defining a leader is to make the natural frequency of a node much higher than * sarika@iiti.ac.in that of other nodes in the network [11,12]. This is one of the traditional way to define a leader in the coupled dynamics on network models. Some other ways of defining leaders are those which depend upon the application and motivation of the problem, e. g., Ref.[13] considers a leader which exchanges information with its neighbors as well as has access to its own state. In Ref.[14] the neuron which fires...

Cluster synchronization of community network with distributed time delays via impulsive control

“…In recent years, synchronization has gained extensive attention from various fields on account of its broad practical applications, such as regulation of power grid, parallel image processing, the operation of no-man air vehicle, the realization of chain detonation, etc. [12,13] Cluster synchronization, [14][15][16][17][18][19][20][21][22] a special synchronization of complex dynamical networks, denotes that synchronization of the nodes in the same community can be achieved, but synchronization of the nodes in different communities cannot be achieved. That is, the nodes in the same or different communities have identical or nonidentical goals.…”

Radial and circular synchronization clusters in extended starlike network of van der Pol oscillators

“…(21) turns to the network used by Bergner et al [33] to study the remote synchronization. Figure 13: Synchronization chart for amplitude equations (21) on the star at L = 2, R = 3, see Fig. 1.…”

“…Network synchronization phenomena are extensively studied [14,15,19,20]. Jalan et al [21] analyze cluster synchronization in presence of a leader. Wang and Chen [22] investigate the control of a scale-free dynamical network by applying local feedback injections to a fraction of network nodes.…”

Variety of regimes of starlike networks of Hénon maps

“…Papers [4,5,9,10] are devoted to an analysis of synchronization conditions, works [11,12] investigate the formation of synchronization clusters while in Ref. [13] the impact of presence of a leader on the cluster synchronization is recovered. Paper [14] investigates so called remote synchronization when nodes can get synchronized even being connected indirectly though intermediate ones.…”