Inertia effects on periodic synchronization in a system of coupled oscillators

We study analytically the synchronization phenomena in a set of globally coupled oscillators under external periodic driving, with emphasis on the effects of small inertia. We examine in detail both the integer and fractional mode locking present in the system with inertia, and derive the self-consistency equation for the order parameter, which reveals variation of the magnitude of the order parameter according to the external periodic driving. In particular, it is found that the inertia induces discontinuous transitions between the coherent and incoherent states.