Origin of Frequency Clusters and Self-Organized Triplet Locking in the Kuramoto Model with Inertia

TL;DR

This paper explores how frequency clusters and triplet locking emerge in the Kuramoto model with inertia, revealing bifurcation mechanisms and the role of homoclinic bifurcations in cluster formation.

Contribution

It identifies the bifurcation processes responsible for creating multiple frequency clusters and triplet locked states in the Kuramoto model with inertia.

Findings

Two frequency clusters are created by homoclinic bifurcations.

Three frequency clusters involve homoclinic and period-doubling bifurcations.

Frequency clusters are not formed by Hopf bifurcations in phase oscillators.

Abstract

We investigate the origin of frequency clusters - states where multiple groups of oscillators with distinct mean frequencies coexist. We use the Kuramoto model with inertia, where identical oscillators are globally coupled. First, we study the creation of two frequency clusters in the thermodynamic limit. Via numerical bifurcation analysis, we confirm that two frequency clusters are created by homoclinic bifurcations. Both clusters can lose their phase-synchrony in transcritical or period-doubling bifurcations. Furthermore, we investigate the creation of three frequency clusters in a system of seven oscillators. Here, the frequency clusters are destabilized by a longitudinal and a transversal period-doubling bifurcation, and the frequency clusters are also created by homoclinic bifurcations. We find that the emergence of three or more frequency clusters via a homoclinic bifurcation implies the creation of a triplet locked state, where the frequency differences exhibit a rational relation. Besides the creation of frequency clusters via a homoclinic bifurcation, we state that Hopf bifurcations cannot create frequency clusters in phase oscillators, and frequency clusters can only be created by global bifurcations.