Comment on "Long Time Evolution of Phase Oscillator Systems" [Chaos 19,023117 (2009), arXiv:0902.2773]

TL;DR

This paper extends previous results on the long-term behavior of large systems of coupled phase oscillators, showing broader conditions under which solutions are attracted to a reduced manifold, thus enhancing the analysis tools for such systems.

Contribution

It generalizes earlier findings by demonstrating attraction to the manifold for more general frequency distributions and less restrictive initial conditions.

Findings

Attraction to the manifold holds for a wider class of frequency distributions.

The initial condition restrictions are significantly relaxed.

The results improve the applicability of the reduced manifold analysis.

Abstract

A previous paper (arXiv:0902.2773, henceforth referred to as I) considered a general class of problems involving the evolution of large systems of globally coupled phase oscillators. It was shown there that, in an appropriate sense, the solutions to these problems are time asymptotically attracted toward a reduced manifold of system states (denoted M). This result has considerable utility in the analysis of these systems, as has been amply demonstrated in recent papers. In this note, we show that the analysis of I can be modified in a simple way that establishes significant extensions of the range of validity of our previous result. In particular, we generalize I in the following ways: (1) attraction to M is now shown for a very general class of oscillator frequency distribution functions g(\omega), and (2) a previous restriction on the allowed class of initial conditions is now substantially relaxed.