Identical phase oscillator networks: bifurcations, symmetry and reversibility for generalized coupling

TL;DR

This paper analyzes the bifurcations, symmetry properties, and reversibility in networks of identical phase oscillators with generalized coupling functions, extending previous bifurcation results and exploring symmetry-induced dynamics.

Contribution

It extends bifurcation analysis to general two harmonic coupling functions and investigates symmetry and reversibility in oscillator networks with various coupling forms.

Findings

For even coupling functions, the system exhibits time-reversal symmetries.

In the case of odd coupling functions, the dynamics depend solely on the coupling function.

The number of constants of motion varies with the harmonic content of the coupling function.

Abstract

For a system of coupled identical phase oscillators with full permutation symmetry, any broken symmetries in dynamical behaviour must come from spontaneous symmetry breaking, i.e. from the nonlinear dynamics of the system. The dynamics of phase differences for such a system depends only on the coupling (phase interaction) function $g(\varphi)$ and the number of oscillators $N$. This paper briefly reviews some results for such systems in the case of general coupling $g$ before exploring two cases in detail: (a) general two harmonic form: $g(\varphi)=q\sin(\varphi-\alpha)+r\sin(2\varphi-\beta)$ and $N$ small (b) the coupling $g$ is odd or even. We extend previously published bifurcation analyses to the general two harmonic case, and show for even $g$ that the dynamics of phase differences has a number of time-reversal symmetries. For the case of even $g$ with one harmonic it is known the system has $N-2$ constants of the motion. This is true for $N=4$ and any $g$, while for $N=4$ and more than two harmonics in $g$, we show the system must have fewer independent constants of the motion.