Synchronization of Coupled Oscillators in a Local One-Dimensional Kuramoto Model

TL;DR

This paper investigates synchronization phenomena in a finite one-dimensional Kuramoto model with local coupling, revealing multiple stable solutions characterized by winding numbers and providing analytical bounds and algorithms for critical coupling.

Contribution

It introduces a modified Kuramoto model on a ring, analyzes multiple stable solutions with winding numbers, and offers an exact calculation method for the critical coupling.

Findings

Finite systems have multiple stable synchronized states.

Phase-locking does not necessarily imply phase coherence in 1D.

An algorithm for calculating the critical coupling is provided.

Abstract

A modified Kuramoto model of synchronization in a finite discrete system of locally coupled oscillators is studied. The model consists of N oscillators with random natural frequencies arranged on a ring. It is shown analytically and numerically that finite-size systems may have many different synchronized stable solutions which are characterised by different values of the winding number. The lower bound for the critical coupling is given, as well as an algorithm for its exact calculation. It is shown that in general phase-locking does not lead to phase coherence in 1D.