Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis

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.

6 pages, 12 figures. Added a reference and corrected typos

References in corpus (7)

• Global phase-locking in finite populations of phase-coupled oscillators
• Phase transitions and topology changes in configuration space
• Enumerating Gribov copies on the lattice
• Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian
• Exploring the energy landscape of XY models
• Enumerating Copies in the First Gribov Region on the Lattice in up to four Dimensions
• Stationary point approach to the phase transition of the classical XY chain with power-law interactions

Cited by in corpus (3)

• Dynamical Systems on Networks: A Tutorial
• Multistability of Phase-Locking in Equal-Frequency Kuramoto Models on Planar Graphs
• Counting equilibria of the Kuramoto model using birationally invariant intersection index