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

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.