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There are no non-zero Stable Fixed Points for dense networks in the homogeneous Kuramoto model
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).Comment: 15 pages 8 figures. arXiv admin note: text overlap with
arXiv:1010.076