Synchronization dynamics in diverse ensemble of noisy phase oscillators with asynchronous phase updates

Decentralized control of autonomous phase oscillators integrated into networked systems is of great interest for many technological applications, from clock synchronization in sensor nets to coordinated motion in swarm robotics. In the simplest distributed synchronization scheme, each oscillator updates its phase from time to time to a new value equal to the average of its present phase and the phases of its neighbors. Here we describe the resulting synchronization dynamics within a mean-field model where the update actions of different oscillators are completely asynchronous. In particular, it is shown how the steady-state level of synchrony depends on noise intensity and frequency diversity for any given rate of updates. The central part of the analysis is devoted to the case when the correction rate positively correlates with the degree of macroscopic coherence. We demonstrate that depending on relation between correction rate and phase coherence the oscillators may exhibit both continuous and discontinuous transition from incoherence to synchrony upon the change of interaction constant. To illustrate our analytical results, numerical simulations have been performed for a large population of phase oscillators with the proposed type of coupling.