  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.
