Synchronisation Conditions In The Kuramoto Model And Their Relationship To Seminorms

In this paper we address two questions about the synchronisation of coupled oscillators in the Kuramoto model with all-to-all coupling. In the first part we use some classical results in convex geometry to prove bounds on the size of the frequency set supporting the existence of stable, phase locked solutions and show that the set of such frequencies can be expressed by a seminorm which we call the Kuramoto norm. In the second part we use some ideas from extreme order statistics to compute upper and lower bounds on the probability of synchronisation for very general frequency distributions. We do so by computing exactly the limiting extreme value distribution of a quantity that is equivalent to the Kuramoto norm.