  We analyze networked heterogeneous nonlinear systems, with diffusive coupling
and interconnected over a generic static directed graph. Due to the network's
hetereogeneity, complete synchronization is impossible, in general, but an
emergent dynamics arises. This may be characterized by two dynamical systems
evolving in two time-scales. The first, "slow", corresponds to the dynamics of
the network on the synchronization manifold. The second, "fast", corresponds to
that of the synchronization errors. We present a framework to analyse the
emergent dynamics based on the behavior of the slow dynamics. Firstly, we give
conditions under which if the slow dynamics admits a globally asymptotically
stable equilibrium, so does the networked systems. Secondly, we give conditions
under which, if the slow dynamics admits an asymptotically stable orbit and a
single unstable equilibrium point, there exists a unique periodic orbit that is
almost-globally asymptotically stable. The emergent behavior is thus clear, the
systems asymptotically synchronize in frequency and, in the limit, as the
coupling strength grows, the emergent dynamics approaches that of the slow
system. Our analysis is established using singular-perturbations theory. In
that regard, we contribute with original statements on stability of
disconnected invariant sets and limit cycles.
