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Algebraic Geometrization of the Kuramoto Model: Equilibri a and Stability Analysis


D. Metha, N. Daleo, F. Dörfler, J. Hauenenstein


Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system of equations, which is an important yet challenging problem. W e translate this into an algebraic geometry problem and use numerical methods to find all of the e quilibria for various choices of coupling constants K , natural frequencies, and on different graphs. We note that f or even modest sizes ( N ∼ 10 − 20 ), the number of equilibria is already more than 100,000. We a nalyze the stability of each computed equilibrium as well as the configu ration of angles. Our exploration of the equilibrium landscape leads to unexpected and possib ly surprising results including non- monotonicity in the number of equilibria, a predictable pat tern in the indices of equilibria, counter- examples to popular conjectures, multi-stable equilibriu m landscapes, scenarios with only unstable equilibria, and multiple distinct extrema in the stable equ ilibrium distribution as a function of the number of cycles in the graph


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% Autogenerated BibTeX entry
@Misc { MetEtal:2014:IFA_5053,
    author={D. Metha and N. Daleo and F. D{\"o}rfler and J. Hauenenstein},
    title={{Algebraic Geometrization of the Kuramoto Model: Equilibri a
	  and Stability Analysis}},
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