# Topological States in the Kuramoto Model

## TL;DR

This paper introduces a novel topological classification of steady states in the Kuramoto model, providing asymptotic estimates for their number and stability as the network size grows.

## Contribution

It develops a new topological approach to classify steady states and derives asymptotic bounds for their quantity and stability in large oscillator networks.

## Key findings

- Derived a Weyl-type asymptotic estimate for steady states.
- Established a lower bound for the number of stable steady states.
- Maximized the asymptotic estimate based on network cycle structure.

## Abstract

The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among the oscillators. In this paper we study steady state solutions of the Kuramoto model by assigning to each steady state a tuple of integers which records how the state twists around the cycles in the network. We then use this new classification of steady states to obtain a "Weyl" type of asymptotic estimate for the number of steady states as the number of oscillators becomes arbitrarily large while preserving the cycle structure. We further show how this asymptotic estimate can be maximized, and as a special case we obtain an asymptotic lower bound for the number of stable steady states of the model.