Time complexity of Synchronization of discrete pulse-coupled oscillators on trees

Abstract

A major open question in the study of synchronization of coupled oscillators is to find necessary and sufficient condition for a system to synchronize on a given family of graphs. This is a difficult question that requires to understand exactly how the nonlienar interaction between local entities evolves over the underlying graph. Another open question is to obtain bounds on the time complexity of synchronization, which has important practical implications in clock synchronization algorithms. We address these questions for one-parameter family of discrete pulse-coupled inhibitory oscillatorscalled the $\kappa$-color firefly cellular automata (FCA). Namely, we show that for $\kappa\le 6$, recurrence of each oscillator is a necessary and sufficient condition for synchronization on finite trees, while for $\kappa \ge 7$ this condition is only necessary. As a corollary, we show that any non-synchronizing dynamics for $\kappa\le 6$ on trees decompose into synchronized subtrees partitioned by `dead' oscillators. Furthermore, on trees with diameter $d$ and maximum degree at most $\kappa$, we show that the worst-case number of iterations until synchronization is of order $O(\kappa d)$ for $\kappa\in \{3,4,5\}$, $O(\kappa d^{2})$ for $\kappa=6$, and infinity for $\kappa \ge 7$. Lastly, we report simulation results of FCA on lattices and conjecture that on a finite square lattice, arbitrary initial configuration is synchronized under $\kappa$-color FCA if and only if $\kappa=4$.