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 Mosc. Math. J., 2010, Volume 10, Number 2, Pages 317–335 (Mi mmj382)

A restricted version of Hilbert's 16th problem for quadratic vector fields

Yu. Ilyashenkoabcd, Jaume Llibree

a Moscow State Universitie, Moscow
b Steklov Math. Institute, Moscow
c Moscow Independent Universitie, Moscow
d Cornell University, US
e Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Catalonia, Spain

Abstract: The restricted version of Hilbert's 16th problem for quadratic vector fields requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector fields considered and the limit cycles to be counted. In this paper we give an upper estimate of the number of limit cycles of quadratic vector fields “$\sigma$-distant from centers and $\kappa$-distant from singular quadratic vector fields” provided that the limit cycles are “$\sigma$-distant from singular points and infinity”.

Key words and phrases: limit cycles, quadratic systems.

DOI: https://doi.org/10.17323/1609-4514-2010-10-2-317-335

Full text: http://www.ams.org/.../abst10-2-2010.html
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MSC: Primary 34C40, 51F14; Secondary 14D05, 14D25
Received: February 4, 2010
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Citation: Yu. Ilyashenko, Jaume Llibre, “A restricted version of Hilbert's 16th problem for quadratic vector fields”, Mosc. Math. J., 10:2 (2010), 317–335

Citation in format AMSBIB
\Bibitem{IlyLli10} \by Yu.~Ilyashenko, Jaume~Llibre \paper A restricted version of Hilbert's 16th problem for quadratic vector fields \jour Mosc. Math.~J. \yr 2010 \vol 10 \issue 2 \pages 317--335 \mathnet{http://mi.mathnet.ru/mmj382} \crossref{https://doi.org/10.17323/1609-4514-2010-10-2-317-335} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2722800} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000279342400003} 

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This publication is cited in the following articles:
1. Peng L., Feng Zh., Liu Ch., “Quadratic Perturbations of a Quadratic Reversible Lotka-Volterra System With Two Centers”, Discret. Contin. Dyn. Syst., 34:11 (2014), 4807–4826
2. Martins R.M., Gomide O.M.L., “Limit Cycles For Quadratic and Cubic Planar Differential Equations Under Polynomial Perturbations of Small Degree”, Discret. Contin. Dyn. Syst., 37:6 (2017), 3353–3386
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