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Synchronization is a universal phenomenon found in many non-equilibriumsystems. Much recent interest in this area has overlapped with the study ofcomplex networks, where a major focus is determining how a system'sconnectivity patterns affect the types of behavior that it can produce. Thusfar, modeling efforts have focused on the tendency of networks of oscillatorsto mutually synchronize themselves, with less emphasis on the effects ofexternal driving. In this work we discuss the interplay between mutual anddriven synchronization in networks of phase oscillators of the Kuramoto type,and explore how the structure and emergence of such states depends on theunderlying network topology for simple random networks with a given degreedistribution. We find a variety of interesting dynamical behaviors, includingbifurcations and bistability patterns that are qualitatively different forheterogeneous and homogeneous networks, and which are separated by aTakens-Bogdanov-Cusp singularity in the parameter region where the couplingstrength between oscillators is weak. Our analysis is connected to theunderlying dynamics of oscillator clusters for important states andtransitions.

Driven synchronization in random networks of oscillators