M\"obius group actions in the solvable chimera model

TL;DR

This paper explores the complex dynamics of oscillator populations in the solvable chimera model using M"obius group actions, revealing subtle behaviors hidden beyond macroscopic observations.

Contribution

It introduces a group-theoretic approach to analyze the detailed dynamics of the chimera model, uncovering hidden variables influencing system behavior.

Findings

Dynamics are more intricate than simple rotations of distributions.

Hidden variables affect the evolution of oscillator densities.

Group actions reveal subtle features not visible at the macroscopic level.

Abstract

We study actions of M\"obius group on two sub-populations in the solvable chimera model proposed by Abrams et al. Dynamics of global variables are given by two coupled Watanabe-Strogatz systems, one for each sub-population. At the first glance, asymptotic dynamics in the model seem to be very simple. For instance, in the stable chimera state distributions of oscillators perform a simple rotations after a certain (sufficiently large) moment. However, a closer look unveils that dynamics are subtler that what can be observed from evolution of densities of oscillators' phases. In order to gain the full picture, one needs to investigate dynamics on the transformation group that acts on these densities. Such an approach emphasizes impact of the "hidden" variable that is not visible on macroscopic level.