Synchronization of discrete oscillators on ring lattices and small-world networks

Abstract

A lattice of three-state stochastic phase-coupled oscillators exhibits a phase transition at a critical value of the coupling parameter a, leading to stable global oscillations. 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 (C3) symmetry is broken. The IP phase does not exist on finite-dimensional lattices. In the case of large negative values of the coupling no synchronization is expected, but nonetheless it was shown that travelling-wave steady states are stable, displaying local order (Escaff et al 2014 Phys. Rev. E 90 052111). 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.