Geometrical Properties of Coupled Oscillators at Synchronization

TL;DR

This paper analytically investigates the synchronization behavior of coupled oscillators in a ring, identifying key oscillators responsible for bifurcations and deriving exact expressions for coupling strength at synchronization.

Contribution

It introduces a geometric and analytical method to determine the coupling strength and identifies oscillators critical to the bifurcation process in synchronized rings.

Findings

Derived an explicit formula for phase differences among oscillators.

Identified oscillators responsible for saddle node bifurcation.

Calculated the exact coupling strength at full synchronization.

Abstract

We study the synchronization of $N$ nearest neighbors coupled oscillators in a ring. We derive an analytic form for the phase difference among neighboring oscillators which shows the dependency on the periodic boundary conditions. At synchronization, we find two distinct quantities which characterize four of the oscillators, two pairs of nearest neighbors, which are at the border of the clusters before total synchronization occurs. These oscillators are responsible for the saddle node bifurcation, of which only two of them have a phase-lock of phase difference equals $\pm$$\pi$/2. Using these properties we build a technique based on geometric properties and numerical observations to arrive to an exact analytic expression for the coupling strength at full synchronization and determine the two oscillators that have a phase-lock condition of $\pm$$\pi$/2.