Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis

TL;DR

This paper applies algebraic geometry and numerical methods to analyze the equilibria and stability of the finite Kuramoto model, revealing complex landscape features and challenging existing conjectures.

Contribution

It introduces an algebraic geometry approach to find all equilibria of the Kuramoto model and uncovers new phenomena in equilibrium configurations and stability patterns.

Findings

Over 100,000 equilibria for N ~ 10-20

Discovery of non-monotonic equilibrium counts

Identification of counter-examples to popular conjectures

Abstract

Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system of equations, which is an important yet challenging problem. We translate this into an algebraic geometry problem and use numerical methods to find all of the equilibria for various choices of coupling constants K, natural frequencies, and on different graphs. We note that for even modest sizes (N ~ 10-20), the number of equilibria is already more than 100,000. We analyze the stability of each computed equilibrium as well as the configuration of angles. Our exploration of the equilibrium landscape leads to unexpected and possibly surprising results including non-monotonicity in the number of equilibria, a predictable pattern in the indices of equilibria, counter-examples to popular conjectures, multi-stable equilibrium landscapes, scenarios with only unstable equilibria, and multiple distinct extrema in the stable equilibrium distribution as a function of the number of cycles in the graph.