There are no non-zero Stable Fixed Points for dense networks in the homogeneous Kuramoto model

TL;DR

This paper proves that dense homogeneous Kuramoto networks with high node degrees have no non-zero stable fixed points, confirming a conjecture about the zero fixed point attracting almost all phase configurations.

Contribution

It develops a necessary condition for stable fixed points and applies it to dense networks, proving the non-existence of non-zero stable fixed points in such cases.

Findings

Dense networks with degree ≥ 0.9395(n-1) lack non-zero stable fixed points.

Confirmed a conjecture that the zero fixed point attracts almost all states in complete networks.

Provided a necessary condition that helps understand stability in Kuramoto models.

Abstract

This paper is concerned with the existence of multiple stable fixed point solutions of the homogeneous Kuramoto model. We develop a necessary condition for the existence of stable fixed points for the general network Kuramoto model. This condition is applied to show that for sufficiently dense n-node networks, with node degrees at least 0.9395(n-1), the homogeneous (equal frequencies) model has no non-zero stable fixed point solution over the full space of phase angles in the range -Pi to Pi. This result together with existing research proves a conjecture of Verwoerd and Mason (2007) that for the complete network and homogeneous model the zero fixed point has a basin of attraction consisting of the entire space minus a set of measure zero. The necessary conditions are also tested to see how close to sufficiency they might be by applying them to a class of regular degree networks studied by Wiley, Strogatz and Girvan (2006).