Impact of a leader on cluster synchronization

We study the mechanisms of frequency-synchronized cluster formation in coupled nonidentical oscillators and investigate the impact of presence of a leader on the cluster synchronization. We find that the introduction of a leader, a 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.

Figure 1

(a) and (b) present variation of $f_{\mathrm{inter}}$ (closed circle), $f_{\mathrm{intra}}$ (open circle), $f_{{\mathrm{N}}_{\mathrm{L}}}$ (fraction of nodes in the largest cluster) (open square) and $F_{\mathrm{clus}}$ (fraction of nodes forming clusters) (open triangle) with increase in $\varepsilon$ and ${\omega }_L$ for the random (a) and scale-free networks (b), (d), (e), (f) of $N=50,\langle k\rangle =2$. (c) presents the variation of the largest Lyapunov exponent ($\lambda$) with change in the coupling strength for the random (solid line) and scale-free (dashed line) networks. $NL,L$, and $PL$ represent the no-leader hub being leader and peripheral node being leader cases, respectively. The figures are obtained by taking average over 20 random initial conditions.

Figure 2

Projection of the phase portraits of coupled R¨ossler oscillators for the no-leader and leader case for a scale-free network of $N=50,\langle k\rangle =2$ at $\varepsilon =2.4$.

Figure 3

Snap shots showing the change in the cluster pattern for random and scale-free networks of $N=50,\langle k\rangle =2$ at $\varepsilon =8$. (a), (d) are plotted for random network and (c), (d) are plotted for scale-free network. The open (red) dots show that the corresponding nodes are synchronized and the closed (black) dots show that the corresponding nodes are connected. The snap shot is plotted after renumbering the nodes so that the nodes forming a cluster come nearby. The star represents the position of the leader in the cluster after rearranging the nodes. $NL$ and $L$ represent the no-leader and leader cases, respectively.

Figure 4

$f_{\mathrm{inter}},f_{\mathrm{intra}}$ and $f_{N_L}$ as a function of ${\omega }_L$ and $\varepsilon$. (a), (b) correspond to the random network and (c), (d) correspond to the 1D lattice of $\langle k\rangle =4$ and $N=50$. All graphs are plotted for average over 20 realizations of the initial conditions.

Figure 5

Snapshots depicting changes in the cluster pattern for scale-free networks with $N=50,\langle k\rangle =2$ at $\varepsilon =3.2$. The natural frequency of the nodes are randomly distributed in the interval $1<\omega <1.09$ and ${\omega }_L=10$.