Multistability of Phase-Locking and Topological Winding Numbers in Locally Coupled Kuramoto Models on Single-Loop Networks

TL;DR

This paper explores the multistability and topological properties of phase-locked solutions in Kuramoto models on networks, revealing how loop currents and winding numbers characterize different stable states and their dependence on coupling strength.

Contribution

It establishes a topological framework linking stable phase-locked solutions to winding numbers and provides algebraic bounds on the number of stable solutions in cycle networks.

Findings

Stable solutions are related by discrete loop currents characterized by winding numbers.

An algebraic upper bound on the number of stable solutions is derived.

Number of stable solutions decreases as coupling strength is lowered.

Abstract

Determining the number of stable phase-locked solutions for locally coupled Kuramoto models is a long-standing mathematical problem with important implications in biology, condensed matter physics and electrical engineering among others. We investigate Kuramoto models on networks with various topologies and show that different phase-locked solutions are related to one another by loop currents. The latter take only discrete values, as they are characterized by topological winding numbers. This result is generically valid for any network, and also applies beyond the Kuramoto model, as long as the coupling between oscillators is antisymmetric in the oscillators' coordinates. Motivated by these results we further investigate loop currents in Kuramoto-like models. We consider loop currents in nonoriented $n$-node cycle networks with nearest-neighbor coupling. Amplifying on earlier works, we give an algebraic upper bound $\mathcal{N} \le 2 \, {\rm Int}[n/4]+1$ for the number $\cal N$ of different, linearly stable phase-locked solutions. We show that the number of different stable solutions monotonically decreases as the coupling strength is decreased. Furthermore stable solutions with a single angle difference exceeding $\pi/2$ emerge as the coupling constant $K$ is reduced, as smooth continuations of solutions with all angle differences smaller than $\pi/2$ at higher $K$. In a cycle network with nearest-neighbor coupling we further show that phase-locked solutions with two or more angle differences larger than $\pi/2$ are all linearly unstable. We point out similarities between loop currents and vortices in superfluids and superconductors as well as persistent currents in superconducting rings and two-dimensional Josephson junction arrays.