Multistability of phase-locking and topological winding numbers in locally coupled Kuramoto models on single-loop networks (Q2798690)

The authors consider the issue of the number of stable phase-locked solutions of the Kuramoto model of sinusoidally coupled phase oscillators. They show that any two different solutions of this form differ by a combination of circulating flows around cycles of the network. These loop flows are quantised and can be labelled by a topological winding number. In the case of a single-cycle network of length \(n\), they show that \(N\), the number of stable solutions, satisfies \(N\leq 2\) \(\mathrm{Int}[n/4]+1\). They also show that a stable solution can have at most one phase difference larger than \(\pi/2\), and that such a solution is directly connected to a solution with all phase differences in \([-\pi/2,\pi/2]\) for the same network and a larger coupling strength.