On mathematical analysis of synchronization to bidirectionally coupled Kuramoto oscillators

Abstract

This article is concerned about the synchronization problem of the bidirectionally coupled Kuramoto model. In this model, each oscillator only interacts with the oscillators with adjacent labeling numbers. Namely, the oscillator theta(i) only interacts with theta(i+1) and theta(i-1). In real applications, this is a typical setting of concatenation connection. We first prove the global convergence of frequency synchronization for the identical oscillators. Also, we present two results of phase synchronization for the identical case in relatively wide initial configuration regimes. In the case that the coupling strength is sufficiently large (equivalent to the difference of the natural frequencies is sufficiently small), we show that for non-identical oscillators within a suitable initial configuration regime, the bidirectionally coupled Kuramoto model achieves a frequency synchronization. The supportive numerical simulations are presented as well. (C) 2020 Elsevier Ltd. All rights reserved.