A Network Topology Dependent Upper Bound on the Number of Equilibria of the Kuramoto Model

We begin with formulating the stationary equations of the Kuramoto model as a system of polynomial equations in a novel way. Then, based on an algebraic geometric root count, we give a prescription of computing an upper bound on the number of equilibria of the Kuramoto model for the most general case, i.e., defined on an arbitrary graph and having generic values of natural frequencies and inhomogeneous couplings. We demonstrate with computational experiments utilizing the numerical polynomial homotopy continuation method that our bound is tight for the number of complex equilibria for the Kuramoto model in the most general case. We then discuss the implications or our results in relation to finding all the real equilibria of the Kuramoto model.