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The study of frequency synchronization configurations in Kuramoto models is aubiquitous mathematical problem that has found applications in many seeminglyindependent fields. In this paper, we focus on networks whose underlying graphare cycle graphs. Based on the recent result on the upper bound of thefrequency synchronization configurations in this context, we propose a toricdeformation homotopy method for locating all frequency synchronizationconfigurations with complexity that is linear in this upper bound. Looselybased on the polyhedral homotopy method, this homotopy induces a deformation ofthe set of the synchronization configurations into a series of toric varieties,yet our method has the distinct advantage of avoiding the costly step ofcomputing mixed cells. We also explore the important consequences of thishomotopy method in the context of direct acyclic decomposition of Kuramotonetworks and tropical stable intersection points for Kuramoto equations.

A toric deformation method for solving Kuramoto equations