Synchronization as Aggregation: Cluster Kinetics of Pulse-Coupled Oscillators

We consider models of identical pulse-coupled oscillators with global interactions. Previous work showed that under certain conditions such systems always end up in sync, but did not quantify how small clusters of synchronized oscillators progressively coalesce into larger ones. Using tools from the study of aggregation phenomena, we obtain exact results for the time-dependent distribution of cluster sizes as the system evolves from disorder to synchrony.

Figure 1

Theoretical and simulated $c(t)$ and $c_1(t)$. Solid lines show theoretical curves obtained analytically (see text). Data points show simulation results for $N={10}^4$ oscillators.

Figure 2

Theoretical and simulated cluster densities $c_2(t)$ through $c_5(t)$. Solid lines show theoretical predictions computed from numerical integration of Eq. (6). Data points show simulation results for $N=5\times {10}^4$ oscillators.