Partially Phase-Locked Solutions to the Kuramoto Model

The Kuramoto model is a canonical model for understanding phase-locking phenomenon. It is well-understood that, in the usual mean-field scaling, full phase-locking is unlikely and that it is partially phase-locked states that are important in applications. Despite this, while there has been much attention given to existence and stability of fully phase-locked states in the finite N Kuramoto model, the partially phase-locked states have received much less attention. In this paper we present two related results. Firstly, we derive an analytical criteria that, for sufficiently strong coupling, guarantees the existence of a partially phase-locked state by proving the existence of an attracting ball around a fixed point of a subset of the oscillators. We also derive a larger invariant ball such that any point in it will asymptotically converge to the attracting ball. Secondly, we consider the large N behavior of the finite N Kuramoto system with randomly distributed frequencies. In the case where the frequencies are independent and identically distributed we use a result of De Smet and Aeyels on partial entrainment to derive a condition giving (with high probability) the existence of a partially entrained cluster. We also derive upper and lower bounds on the size of the largest entrained cluster, together with a lower bound on the order parameter. Interestingly in a series of numerical experiments we find that the observed size of the largest entrained cluster is predicted extremely well by the upper bound.