Synchronization of Discrete Oscillators on Ring Lattices and Small-World Networks

TL;DR

This paper investigates how discrete oscillators on ring and small-world networks synchronize or form traveling waves, revealing phase transitions and metastable states influenced by network connectivity and initial conditions.

Contribution

It extends understanding of oscillator synchronization by analyzing long-range coupling effects and metastable states beyond complete graphs.

Findings

Global oscillations occur at a critical coupling value.

Infinite-period transition leads to broken symmetry.

Traveling-wave states are metastable and depend on initial conditions.

Abstract

A lattice of three-state stochastic phase-coupled oscillators introduced by Wood it et al. exhibits a phase transition at a critical value of the coupling parameter $a$, leading to stable global oscillations (GO). On a complete graph, upon further increase in $a$, the model exhibits an infinite-period (IP) phase transition, at which collective oscillations cease and discrete rotational ($C_3$) symmetry is broken. In the case of large negative values of the coupling, Escaff et al. discovered the stability of travelling-wave states with no global synchronization but with local order. Here, we verify the IP phase in systems with long-range coupling but of lower connectivity than a complete graph and show that even for large positive coupling, the system sometimes fails to reach global order. The ensuing travelling-wave state appears to be a metastable configuration whose birth and decay (into the previously described phases) are associated with the initial conditions and fluctuations.