Long Time Evolution of Phase Oscillator Systems

TL;DR

This paper proves that the long-term behavior of large, globally coupled phase oscillator systems can be fully described by a known reduced manifold, simplifying the analysis of their attractors and bifurcations.

Contribution

It establishes that the reduced manifold captures all long-term dynamics of the system's order parameters, confirming its completeness for describing attractors.

Findings

Attractors of the order parameter dynamics are the only attractors of the full system.

Long-term behavior can be fully described by the reduced manifold.

The result simplifies analysis of bifurcations and attractors in large oscillator systems.

Abstract

It is shown, under weak conditions, that the dynamical evolution of an important class of large systems of globally coupled, heterogeneous frequency, phase oscillators is, in an appropriate physical sense, time-asymptotically attracted toward a reduced manifold of system states. This manifold, which is invariant under the system evolution, was previously known and used to facilitate the discovery of attractors and bifurcations of such systems. The result of this paper establishes that attractors for the order parameter dynamics obtained by restriction to this reduced manifold are, in fact, the only such attractors of the full system. Thus all long time dynamical behavior of the order parameters of these systems can be obtained by restriction to the reduced manifold.