Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators

TL;DR

This paper investigates how breaking symmetry in a minimal four-oscillator network leads to the emergence of weak chimera states, including chaotic dynamics, through bifurcations and global bifurcation scenarios.

Contribution

It provides a detailed analysis of symmetry-breaking bifurcations in the smallest network supporting chimera states and chaos in Kuramoto-type oscillators.

Findings

Symmetry breaking enables frequency-unlocked chimera solutions.

Weak chimeras can undergo period doubling and chaos.

Smallest network supporting chaos is four oscillators.

Abstract

Despite their simplicity, networks of coupled phase oscillators can give rise to intriguing collective dynamical phenomena. However, the symmetries of globally and identically coupled identical units do not allow solutions where distinct oscillators are frequency-unlocked -- a necessary condition for the emergence of chimeras. Thus, forced symmetry breaking is necessary to observe chimera-type solutions. Here, we consider the bifurcations that arise when full permutational symmetry is broken for the network to consist of coupled populations. We consider the smallest possible network composed of four phase oscillators and elucidate the phase space structure, (partial) integrability for some parameter values, and how the bifurcations away from full symmetry lead to frequency-unlocked weak chimera solutions. Since such solutions wind around a torus they must arise in a global bifurcation scenario. Moreover, periodic weak chimeras undergo a period doubling cascade leading to chaos. The resulting chaotic dynamics with distinct frequencies do not rely on amplitude variation and arise in the smallest networks that support chaos.