Partial entrainment in the finite Kuramoto–Sakaguchi model

Abstract

Although modifications of the Kuramoto model have been the subject of extensive research, the model itself is not yet fully understood. We offer several results and observations, some analytic, others through simulations. We derive a sufficient condition for the existence of a solution exhibiting partial entrainment with respect to a given subset of oscillators; the result also implies persistence of the entrainment behavior under perturbations. The critical values of the coupling strength, defining the transitions between different forms of partial entrainment, are predicted by an analytical approximation, based on the fact that oscillators with large differences in their natural frequencies have little influence on each other’s entrainment behavior; the predictions agree with the actual values, obtained by simulations. We indicate (by simulations) that entrainment can disappear with increasing coupling strength, and that, in arrays of Josephson junctions, a similar phenomenon can be observed, where it is also possible that a junction leaving one entrained subset joins another entrained subset.