Double Transition to Synchronization: A Generic Emergent Transitional Behavior in Large Systems of Coupled Oscillators

We study a system of M ≫ 1 nonidentical laser oscillators coupled with time delay through a central laser oscillator in a star-coupling topology. In the large-M limit we show that this system undergoes novel transitional behavior which represents a new generic type of emergent behavior in large systems of coupled laser oscillators. Specifically, we observe a sequence of two dynamical transitions ("a double transition"). A first transition occurs when the effective threshold of the central oscillators is reached leading to an increase of the order parameter to a level far above the noise level. As the control parameter is raised further a second transition occurs leading to a further increase of the order parameter to a level of order M^1/2 above the level achieved past the first transition. A scaling analysis and numerical experiments reveal the underlying mechanism of this scenario. We suggest that double transitions, discovered here in the context of a specific system, are expected to occur in diverse situations involving large coupled-oscillator networks.