Global bifurcations organizing weak chimeras in three symmetrically coupled Kuramoto oscillators with inertia

TL;DR

This paper investigates the bifurcation structures leading to weak chimeras in three coupled Kuramoto oscillators with inertia, revealing symmetry-breaking heteroclinic bifurcations as organizing centers for these phenomena.

Contribution

It identifies and characterizes specific symmetry-related bifurcations that organize the emergence of weak chimeras in a minimal oscillator system.

Findings

Identified a non-transverse heteroclinic bifurcation as an organizing center.

Discovered a second organizing center related to symmetry-breaking heteroclinic connections.

Showed how bifurcations lead to the formation of weak chimeras in the system.

Abstract

Frequency desynchronized attractors cannot appear in identically coupled symmetric phase oscillators because "overtaking" of phases cannot occur. This restriction no longer applies for more general identically coupled oscillators. Hence, it is interesting to understand precisely how frequency synchrony is lost and how invariant sets such as attracting weak chimeras are generated at torus breakup, where the phase description breaks down. Maistrenko et al (2016) found numerical evidence of an organizing center for weak chimeras in a system of $N=3$ coupled identical Kuramoto oscillators with inertia. This paper identifies this organizing center and shows that it corresponds to a particular type of non-transverse heteroclinic bifurcation that is generic in the context of symmetry. At this codimension two bifurcation there is a splitting of connecting orbits between the in-phase (fully synchronized) state. This generates a wide variety of associated bifurcations to weak chimeras. We further highlight a second organizing center associated with a codimension two symmetry-breaking heteroclinic connection.