  Previous results have shown that a large class of complex systems consisting
of many interacting heterogeneous phase oscillators exhibit an attracting
invariant manifold. This result has enabled reduced analytic system
descriptions from which all the long term dynamics of these systems can be
calculated. Although very useful, these previous results are limited by the
restriction that the individual interacting system components have
one-dimensional dynamics, with states described by a single, scalar, angle-like
variable (e.g., the Kuramoto model). In this paper we consider a generalization
to an appropriate class of coupled agents with higher-dimensional dynamics. For
this generalized class of model systems we demonstrate that the dynamics again
contain an invariant manifold, hence enabling previously inaccessible analysis
and improved numerical study, allowing a similar simplified description of
these systems. We also discuss examples illustrating the potential utility of
our results for a wide range of interesting situations.
