Mathematics – Dynamical Systems
Scientific paper
2011-09-21
Mathematics
Dynamical Systems
15 pages 8 figures. arXiv admin note: text overlap with arXiv:1010.0766
Scientific paper
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).
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