Statistics of synchronization times in Kuramoto oscillators

Abstract

The synchronization of coupled systems is a widely studied phenomenon in nonlinear science. As a completely synchronized state emerges asymptotically, it is pertinent to quantify the statistics of the timescales in which it is attained. We study the Kuramoto model, a paradigmatic model of synchronization, and record the time at which the synchronized state is reached for the first time. The First Synchronization Time (FST), on suitable rescaling, exhibits a universal distribution for a wide range of parameter values and even in the presence of noise. We obtain an approximate time evolution of the order parameter using the celebrated Ott-Antonsen ansatz and analytically demonstrate that the FST exhibits a universal Gumbel distribution.