On the typical and atypical solutions to the Kuramoto equations

TL;DR

This paper investigates the solutions of the Kuramoto equations by relating their equilibrium count to network combinatorics, providing bounds, conditions for strictness, and examples of infinite equilibria.

Contribution

It formulates the Kuramoto equations algebraically, connects equilibrium counts to the adjacency polytope volume, and identifies conditions for infinite solutions.

Findings

Complex root count equals normalized volume of adjacency polytope

Explicit algebraic conditions for strict bounds

Existence of networks with infinitely many equilibria

Abstract

The Kuramoto model is a dynamical system that models the interaction of coupled oscillators. There has been much work to effectively bound the number of equilibria to the Kuramoto model for a given network. By formulating the Kuramoto equations as a system of algebraic equations, we first relate the complex root count of the Kuramoto equations to the combinatorics of the underlying network by showing that the complex root count is generically equal to the normalized volume of the corresponding adjacency polytope of the network. We then give explicit algebraic conditions under which this bound is strict and show that there are networks where the Kuramoto equations have infinitely many equilibria.