On the root count of algebraic Kuramoto equations in cycle networks with uniform coupling

TL;DR

This paper refines upper bounds on the number of synchronization solutions in cycle networks of oscillators with uniform coupling, revealing a special case when the number of oscillators is divisible by 4.

Contribution

It provides a precise analysis of the root count for algebraic Kuramoto equations in uniform coupling cycle networks, highlighting a unique divisibility condition.

Findings

Upper bounds are lower when N is divisible by 4.

Explicit formula for the gap between algebraic bounds.

Identifies a special case with reduced solution count.

Abstract

The Kuramoto model is a classical model used in the study of spontaneous synchronizations in networks of coupled oscillators. In this model, frequency synchronization configurations can be formulated as complex solutions to a system of algebraic equations. Recently, upper bounds to the number of frequency synchronization configurations in cycle networks of N oscillators were calculated under the assumption of generic non-uniform coupling. In this paper, we refine these results for the special cases of uniform coupling. In particular, we show that when, and only when, N is divisible by 4, the upper bound for the number of synchronization configurations in the uniform coupling cases is significantly less than the bound in the non-uniform coupling cases. This result also establishes an explicit formula for the gap between the birationally invariant intersection index and the Bernshtein-Kushnirenko-Khovanskii bound for the underlying algebraic equations.