Hopf Bifurcations of Twisted States in Phase Oscillators Rings with Nonpairwise Higher-Order Interactions

TL;DR

This paper investigates how nonpairwise higher-order interactions in ring networks of phase oscillators lead to Hopf bifurcations of twisted states, resulting in quasiperiodic solutions that can be stable, expanding understanding of synchronization phenomena.

Contribution

It analyzes Hopf bifurcations of twisted states in oscillator rings with higher-order interactions, revealing new stable quasiperiodic solutions and their relation to anti-phase states.

Findings

Hopf bifurcations produce stable quasiperiodic solutions.

Emergent solutions approach anti-phase and different twisted states.

Higher-order interactions enable stability of complex oscillatory patterns.

Abstract

Synchronization is an essential collective phenomenon in networks of interacting oscillators. Twisted states are rotating wave solutions in ring networks where the oscillator phases wrap around the circle in a linear fashion. Here, we analyze Hopf bifurcations of twisted states in ring networks of phase oscillators with nonpairwise higher-order interactions. Hopf bifurcations give rise to quasiperiodic solutions that move along the oscillator ring at nontrivial speed. Because of the higher-order interactions, these emerging solutions may be stable. Using the Ott--Antonsen approach, we continue the emergent solution branches which approach anti-phase type solutions (where oscillators form two clusters whose phase is $\pi$ apart) as well as twisted states with a different winding number.