  Synchronization is a universal phenomenon found in many non-equilibrium
systems. Much recent interest in this area has overlapped with the study of
complex networks, where a major focus is determining how a system's
connectivity patterns affect the types of behavior that it can produce. Thus
far, modeling efforts have focused on the tendency of networks of oscillators
to mutually synchronize themselves, with less emphasis on the effects of
external driving. In this work we discuss the interplay between mutual and
driven synchronization in networks of phase oscillators of the Kuramoto type,
and explore how the structure and emergence of such states depends on the
underlying network topology for simple random networks with a given degree
distribution. We find a variety of interesting dynamical behaviors, including
bifurcations and bistability patterns that are qualitatively different for
heterogeneous and homogeneous networks, and which are separated by a
Takens-Bogdanov-Cusp singularity in the parameter region where the coupling
strength between oscillators is weak. Our analysis is connected to the
underlying dynamics of oscillator clusters for important states and
transitions.

Adaptation and Self-Organizing Systems;Chaotic Dynamics;Physics and Society