Connecting minimal chimeras and fully asymmetric chaotic attractors through equivariant pitchfork bifurcations

TL;DR

This paper explores how symmetric networks of four coupled oscillators can exhibit complex symmetry-broken states, including minimal chimeras and asymmetric chaotic attractors, through a series of bifurcations.

Contribution

It uncovers the interconnected bifurcation structure linking minimal chimeras and asymmetric chaotic states via equivariant pitchfork bifurcations in symmetric oscillator networks.

Findings

Identification of chaotic 2-1-1 minimal chimeras from period-doubling cascades.

Discovery of fully asymmetric chaotic states arising from periodic solutions.

Bifurcation structure connecting different symmetry-broken states through pitchfork bifurcations.

Abstract

Highly symmetric networks can exhibit partly symmetry-broken states, including clusters and chimera states, i.e., states of coexisting synchronized and unsynchronized elements. We address the $\mathbb{S}_4$ permutation symmetry of four globally coupled Stuart-Landau oscillators and uncover an interconnected web of differently symmetric solutions. Among these are chaotic $2\!-\!1\!-\!1$ minimal chimeras that arise from $2\!-\!1\!-\!1$ periodic solutions in a period-doubling cascade, as well as fully asymmetric chaotic states arising similarly from periodic $1\!-\!1\!-\!1\!-\!1$ solutions. A backbone of equivariant pitchfork bifurcations mediate between the two cascades, culminating in equivariant pitchforks of chaotic attractors.