Single-clustering synchronization in a ring of Kuramoto oscillators

Abstract

Multiple-clustering synchronization is a common scenario of global phase synchronization. However, a novel single-clustering type of phase synchronization in a ring of Kuramoto oscillators has been recently reported in studying the influence of the permutation of the natural frequencies of oscillators on the synchronization efficiency (Wu et al 2012 Europhys. Lett. 97 40005). It was found that it occurs for a particular spatial frequency distribution and gives rise to a very small critical coupling strength Kc even if the oscillator number is large. Here we focus on this particular type of synchronization and study its generality. We provide some solid evidence for the convergence of Kc to a small constant in the thermodynamic limit, based on the finite size analysis. Further we demonstrate that it is robust in the sense of either switching the natural frequencies of any two oscillators or randomly perturbing the frequencies of all coupled oscillators. All these findings prove that the single-clustering synchronization is indeed generically observable, with merit for potential engineering applications.