Interplay among inertia, time delay, and frustration on synchronization dynamics

The interplay between time delay and phase shifts plays an important role for the regulation of synchronization and gives rise to interesting collective dynamical phenomena. Here we derive a stability criterion for the synchronous state in a system of identical oscillators in the presence of inertia, time delay, and a phase shift. We investigate how the combination of these factors influences the synchronization rates of the system approaching steady states after small perturbations. In particular, we prove that the global synchronization rate is twice the smallest local synchronization rate. We also find that the fastest synchronization rate increases with inertia but occurs at smaller values of critical delay.