Conformists and contrarians in a Kuramoto model with identical natural frequencies

TL;DR

This paper explores a modified Kuramoto model with identical natural frequencies, incorporating both conformist and contrarian oscillators, revealing complex dynamics including faction splitting, traveling waves, and incoherence, analyzed through exact mathematical solutions.

Contribution

It introduces a novel variant of the Kuramoto model with negative coupling oscillators and provides exact solutions for its diverse dynamical states.

Findings

System exhibits faction splitting and traveling waves.

Exact bifurcation analysis of different states.

Contrarian oscillators induce richer dynamics than heterogeneous models.

Abstract

We consider a variant of the Kuramoto model, in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field. These "contrarian" oscillators tend to align in antiphase with the mean field, whereas the positively coupled "conformist" oscillators favor an in-phase relationship. The interplay between these effects can lead to rich dynamics. In addition to a splitting of the population into two diametrically opposed factions, the system can also display traveling waves, complete incoherence, and a blurred version of the two-faction state. Exact solutions for these states and their bifurcations are obtained by means of the Watanabe-Strogatz transformation and the Ott-Antonsen ansatz. Curiously, this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies.