Long Time Evolution of Phase Oscillator Systems

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.