Symmetry and symmetry breaking in coupled oscillator communities

TL;DR

This paper investigates how symmetry and symmetry-breaking influence the collective dynamics of coupled oscillator communities, revealing complex bifurcation structures and new states when symmetry is broken.

Contribution

It provides a comparative analysis of symmetric and asymmetric coupled oscillator systems, highlighting the effects of symmetry-breaking on bifurcation diagrams and state transitions.

Findings

Symmetric systems exhibit bifurcations between incoherence, standing waves, and partial synchronization.

Symmetry-breaking introduces new bifurcations, such as Hopf bifurcations, and additional synchronized states.

Bistability regions are expanded or altered due to symmetry-breaking.

Abstract

With the recent development of analytical methods for studying the collective dynamics of coupled oscillator systems, the dynamics of communities of coupled oscillators have received a great deal of attention in the nonlinear dynamics community. However, the majority of these works treat systems with a number of symmetries to simplify the analysis. In this work we study the role of symmetry and symmetry-breaking in the collective dynamics of coupled oscillator communities, allowing for a comparison between the macroscopic dynamics of symmetric and asymmetric systems. We begin by treating the symmetric case, deriving the bifurcation diagram as a function of intra- and inter-community coupling strengths. In particular we describe transitions between incoherence, standing wave, and partially synchronized states and reveal bistability regions. When we turn our attention to the asymmetric case we find that the symmetry-breaking complicates the bifurcation diagram. For instance, a pitchfork bifurcation in the symmetric case is broken, giving rise to a Hopf bifurcation. Moreover, an additional partially synchronized state emerges, as well as a new bistability region.