Analytic Determination of the Critical Coupling for Oscillators in a Ring

TL;DR

This paper analytically determines the critical coupling strength needed for complete synchronization in a ring of coupled oscillators with nearest-neighbor interactions, revealing dependence on initial frequencies.

Contribution

It introduces an analytical method to find the critical coupling for synchronization in a ring oscillator model based on boundary oscillator analysis.

Findings

Identifies boundary oscillators that determine phase-locking.

Provides an explicit formula for the critical coupling.

Shows the critical coupling depends only on initial frequencies.

Abstract

We study a model of coupled oscillators with bidirectional first nearest neighbours coupling with periodic boundary conditions. We show that a stable phase-locked solution is decided by the oscillators at the borders between the major clusters, which merge to form a larger one of all oscillators at the stage of complete synchronization. We are able to locate these four oscillators as well as the size of major clusters in the vicinity of the stage of full synchronization which we show to depend only on the set of initial frequencies. Using the method presented here, we are able to obtain an analytic form of the critical coupling, at which the complete synchronization state occurs.