Hierarchy of exact low-dimensional reductions for populations of coupled oscillators

TL;DR

This paper introduces a hierarchy of exact low-dimensional models for populations of coupled oscillators, enabling analysis of complex dynamics like chaos through an extended manifold approach.

Contribution

It proposes an exact ansatz for the phase distribution's moments, generalizing the Ott-Antonsen reduction to higher dimensions and complex behaviors.

Findings

Exact truncation of the moment hierarchy at arbitrary modes

Demonstration of chaos in oscillator populations using extended manifolds

Generalization to oscillators with Cauchy-Lorentzian frequency distribution

Abstract

We consider an ensemble of phase oscillators in the thermodynamic limit, where it is described by a kinetic equation for the phase distribution density. We propose an ansatz for the circular moments of the distribution (Kuramoto-Daido order parameters) that allows for an exact truncation at an arbitrary number of modes. In the simplest case of one mode, the ansatz coincides with that of Ott and Antonsen [Chaos 18, 037113 (2008)]. Dynamics on the extended manifolds facilitate higher dimensional behavior such as chaos, which we demonstrate with a simulation of a Josephson junction array. The findings are generalized for oscillators with a Cauchy-Lorentzian distribution of natural frequencies.