Global synchronization of pulse-coupled oscillators on trees

Abstract

Consider a distributed network on a finite simple graph $G=(V,E)$ with diameter $d$ and maximum degree $Δ$, where each node has a phase oscillator revolving on $S^{1}=\mathbb{R}/\mathbb{Z}$ with unit speed. Pulse-coupling is a class of distributed time evolution rule for such networked phase oscillators inspired by biological oscillators, which depends only upon event-triggered local pulse communications. In this paper, we propose a novel inhibitory pulse-coupling and prove that arbitrary phase configuration on $G$ synchronizes by time $51d$ if $G$ is a tree and $Δ\le 3$. We extend this pulse-coupling by letting each oscillator throttle the input according to an auxiliary state variable. We show that the resulting adaptive pulse-coupling synchronizes arbitrary initial configuration on $G$ by time $83d$ if $G$ is a tree. As an application, we obtain a universal randomized distributed clock synchronization algorithm, which uses $O(\log Δ)$ memory per node and converges on any $G$ with expected worst case running time of $O(|V|+(d^{5}+Δ^{2})\log |V|)$.

41 pages, 21 figures, preprint. Definition of the adaptive 4-coupling is presented as a pesudocode. Simulation of the adaptive 4-coupling modulo $M=64$ is added in Figure 1, SIAM Journal on Applied Dynamical Systems, 2018