Clusterization and phase diagram of the bimodal Kuramoto model with bounded confidence

TL;DR

This paper extends the Kuramoto model by incorporating confidence bounds, analyzing how oscillators form clusters based on phase differences and bimodal frequency distributions, and constructs a phase diagram for these fixed points.

Contribution

It introduces a bounded confidence mechanism into the Kuramoto model and analytically characterizes the resulting clustering and phase diagram for bimodal frequency distributions.

Findings

Fixed points are composed of independent oscillator clusters.

Cluster formation depends on confidence bound length and frequency peak separation.

The phase diagram of attractive fixed points is analytically derived.

Abstract

Inspired by the Deffuant and Hegselmann-Krause models of opinion dynamics, we extend the Kuramoto model to account for confidence bounds, i.e., vanishing interactions between pairs of oscillators when their phases differ by more than a certain value. We focus on Kuramoto oscillators with peaked, bimodal distribution of natural frequencies. We show that, in this case, the fixed-points for the extended model are made of certain numbers of independent clusters of oscillators, depending on the length of the confidence bound -- the interaction range -- and the distance between the two peaks of the bimodal distribution of natural frequencies. This allows us to construct the phase diagram of attractive fixed-points for the bimodal Kuramoto model with bounded confidence and to analytically explain clusterization in dynamical systems with bounded confidence.