Isotropy of Angular Frequencies and Weak Chimeras With Broken Symmetry

TL;DR

This paper extends the concept of weak chimeras in oscillator networks by linking solution isotropy to angular frequency vectors, demonstrating the existence of symmetry-breaking weak chimeras and chaotic states in coupled phase oscillators.

Contribution

It generalizes weak chimera definitions to broader dynamical systems using isotropy of angular frequencies and constructs examples of symmetry-breaking and chaotic weak chimeras.

Findings

Weak chimeras can exist without symmetries in solutions.

A coupling function can induce chaotic weak chimeras.

Symmetries imply frequency synchronization, but are not necessary.

Abstract

The notion of a weak chimeras provides a tractable definition for chimera states in networks of finitely many phase oscillators. Here we generalize the definition of a weak chimera to a more general class of equivariant dynamical systems by characterizing solutions in terms of the isotropy of their angular frequency vector - for coupled phase oscillators the angular frequency vector is given by the average of the vector field along a trajectory. Symmetries of solutions automatically imply angular frequency synchronization. We show that the presence of such symmetries is not necessary by giving a result for the existence of weak chimeras without instantaneous or setwise symmetries for coupled phase oscillators. Moreover, we construct a coupling function that gives rise to chaotic weak chimeras without symmetry in weakly coupled populations of phase oscillators with generalized coupling.