A toric deformation method for solving Kuramoto equations on cycle networks

The study of frequency synchronization configurations in Kuramoto models for networks of coupled oscillators is a ubiquitous mathematical problem that has found applications in many seemingly independent fields. In this paper, we focus on networks in which the underlying graph is a cycle graph. Based on a recent result on the maximum number of distinct frequency synchronization configurations in this context, we propose a constructive toric deformation homotopy method for locating all frequency synchronization configurations with complexity that is linear in this upper bound. Inspired by the polyhedral homotopy method for solving general polynomial systems and the more general framework of toric deformation in algebraic geometry, the proposed homotopy method deforms the set of synchronization configurations into a collection of toric varieties. Compared to existing homotopy methods for solving Kuramoto equations, the proposed method has the distinct advantages of avoiding the costly step of computing mixed volume/cells and using special starting systems that can be solved in linear time. We also explore the important consequences of this homotopy method in the context of directed acyclic decompositions of Kuramoto networks and tropical stable intersection points for Kuramoto equations.

Automated summary

The study focuses on frequency synchronization configurations in Kuramoto models for networks of coupled oscillators with a cycle graph as the underlying graph. A constructive toric deformation homotopy method is proposed to locate all frequency synchronization configurations with linear complexity. The proposed method has advantages over existing homotopy methods in terms of computational cost and solving time.