Arnold tongues in the forced Kuramoto model with matrix coupling

TL;DR

This paper explores how matrix coupling in a generalized Kuramoto model leads to multiple resonance phenomena, including Arnold tongues and devil's staircases, under external periodic forcing, expanding understanding beyond the classic model.

Contribution

It introduces a matrix-coupled Kuramoto model with external forcing and demonstrates the emergence of multiple Arnold tongues through numerical simulations.

Findings

Multiple Arnold tongues observed with matrix coupling.

Resonant modes include complex devil's staircase structures.

Contrasts with single 1:1 resonance in original model.

Abstract

We consider a generalization of the Kuramoto model in which phase oscillators are represented by unit vectors coupled by a matrix of constant coefficients. We show that, when the oscillators are driven by an external periodic force, several resonances appear, giving rise to Arnold tongues that can be observed as the intensity and frequency of the external force are varied. Applying the Ott-Antonsen ansatz we obtain equations for the module and phase of order parameter. As these equations are explicitly time-dependent, we resort to extensive numerical simulations to uncover the resonant modes and their associated Arnold tongues and devil's staircases. These results contrast with the original forced Kuramoto model, where only $1:1$ resonance is possible.