  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.
