Synchronization of coupled second-order Kuramoto-Sakaguchi oscillators

We study the synchronization of oscillators with inertias and phase shifts, namely the second-order Kuramoto-Sakaguchi model. Using the self-consistent method, we find that the effect of inertia is the introduction of effective phase shifts. The discontinuous synchronization transition of the Kuramoto-Sakaguchi model changes to a continuous one when the value of inertia is small. In addition, we find a new synchronization process, in which with increasing coupling strength the system reaches an oscillating state instead of complete synchronization due to the cross-effect of phase shifts and inertias. Through numerical simulations, the same type of synchronization process is also found for oscillators in complex networks, including scale-free, small-world and random networks.