Dynamics on invariant tori emerging through forced symmetry breaking in phase oscillator networks

TL;DR

This paper investigates how invariant tori in coupled phase oscillator networks deform under forced symmetry breaking, revealing new bifurcating stable synchrony patterns through analytical and numerical methods.

Contribution

It introduces a parametrization method to analyze perturbed invariant tori and applies it to uncover new bifurcating synchrony patterns in oscillator networks.

Findings

Perturbed tori can be computed using asymptotic expansion techniques.

Numerical bifurcation analysis reveals new stable synchrony patterns.

These patterns can serve as building blocks for complex network dynamics.

Abstract

We consider synchrony patterns in coupled phase oscillator networks that correspond to invariant tori. For specific nongeneric coupling, these tori are equilibria relative to a continuous symmetry action. We analyze how the invariant tori deform under forced symmetry breaking as more general network interaction terms are introduced. We first show in general that perturbed tori that are relative equilibria can be computed using a parametrization method; this yields an asymptotic expansion of an embedding of the perturbed torus, as well as the local dynamics on the torus. We then apply this result to a coupled oscillator network, and we numerically study the dynamics on the persisting tori in the network by looking for bifurcations of their periodic orbits in a boundary-value-problem setup. This way we find new bifurcating stable synchrony patterns that can be the building blocks of larger global structures such as heteroclinic cycles.