Spontaneous phase oscillation induced by inertia and time delay

We consider a system of coupled oscillators with finite inertia and time-delayed interaction, and investigate the interplay between inertia and delay both analytically and numerically. The phase velocity of the system is examined: revealed in numerical simulations is the emergence of spontaneous phase oscillation without external driving, which turns out to be in good agreement with analytical results derived in the strong-coupling limit. Such self-oscillation is found to Suppress synchronization, and its frequency is observed to decrease with inertia and delay. We obtain the phase diagram, which displays oscillatory and stationary phases in the appropriate regions of the parameters.