Phase oscillators with global sinusoidal coupling evolve by Mobius group action

TL;DR

This paper reveals that the low-dimensional dynamics of identical phase oscillators with sinusoidal coupling are governed by the Mobius group, which explains their invariant manifolds and constants of motion, and relates to observed chaotic behaviors.

Contribution

It proves that the equations of such oscillators are generated by Mobius group actions, providing a geometric and algebraic explanation for their low-dimensional dynamics.

Findings

The governing equations are generated by Mobius group actions.

The state space is partitioned into three-dimensional invariant manifolds.

Invariant manifolds often contain regions of neutrally stable chaos.

Abstract

Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems, by proving that the governing equations are generated by the action of the Mobius group, a three-parameter subgroup of fractional linear transformations that map the unit disc to itself. When there are no auxiliary state variables, the group action partitions the N-dimensional state space into three-dimensional invariant manifolds (the group orbits). The N-3 constants of motion associated with this foliation are the N-3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos.