Explosive synchronization enhanced by time-delayed coupling

This paper deals with the emergence of explosive synchronization in scale-free networks by considering the Kuramoto model of coupled phase oscillators. The natural frequencies of oscillators are assumed to be correlated with their degrees, and a time delay is included in the system. This assumption allows enhancing the explosive transition to reach a synchronous state. We provide an analytical treatment developed in a star graph, which reproduces results obtained in scale-free networks. Our findings have important implications in understanding the synchronization of complex networks since the time delay is present in most real-world complex systems due to the finite speed of the signal transmission over a distance.

Figure 1

Coherence diagram $r\left(\lambda \right)$ for the BA model with $N=1000$ and $\langle k\rangle =6$.

Figure 2

Coherence diagram $r\left(\lambda \right)$ for a star network composed of (a) $K=60$ and (b) $K=80$ leaves.

Figure 3

Average frequency $\Omega$ as a function of time delay $\tau$ for star networks composed of (a) $K=60$ and (b) $K=80$ leaves.

Figure 4

Coherence diagram $r\left(\lambda \right)$ for star networks composed of (a) $K=60$ and (b) $K=80$ leaves.

Figure 5

Comparison of the explosive synchronization observed in star networks with sizes $K=60$, 70, 80, and 90 for the time delay of $\tau =2.4$. The inset shows the same star networks but for the case with $\tau =0$.

Figure 6

Theoretical and numerical curves for the frequency $\Omega$ as a function of $\tau$ with $\lambda =1.0$ for the cases (a) $K=60$ and (b) $K=80$. The discrete points correspond to the data shown in Fig. 3.

Figure 7

Synchronization diagrams for a star network with $K=80$ leaves and theoretical estimation for the critical coupling ${\lambda }_c$ for each time delay displayed.