The Kuramoto model of coupled oscillators with a bi-harmonic coupling function

TL;DR

This paper analyzes synchronization phenomena in a Kuramoto model with bi-harmonic coupling, providing analytical solutions for order parameters and revealing coexistence of synchronous and asynchronous states with finite lifetime.

Contribution

It introduces an analytical method for solving self-consistent equations in a bi-harmonic Kuramoto model, exploring both single- and multi-branch entrainment states.

Findings

Synchronous states coexist with neutrally stable asynchronous regimes.

Asynchronous states have a finite lifetime that grows with ensemble size.

Analytical solutions describe the complex order parameters in large populations.

Abstract

We study synchronization in a Kuramoto model of globally coupled phase oscillators with a bi-harmonic coupling function, in the thermodynamic limit of large populations. We develop a method for an analytic solution of self-consistent equations describing uniformly rotating complex order parameters, both for single-branch (one possible state of locked oscillators) and multi-branch (two possible values of locked phases) entrainment. We show that synchronous states coexist with the neutrally linearly stable asynchronous regime. The latter has a finite life time for finite ensembles, this time grows with the ensemble size as a power law.