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