Quadratic perturbations of a quadratic reversible center of genus one
type: Item
instance of: scholarly article
title: Quadratic perturbations of a quadratic reversible center of genus one
zbMATH Open document ID: 1232.34050
author: Linping Peng
published in: Frontiers of Mathematics in China
publication date: 2011-11-04T00:00:00Z
review text: The bifurcation of limit cycles for small quadratic perturbations of the planar system with a center \[ x'=y-28x^2+32y^2,\quad y'=-x(1+32y) \] is studied. This is case (r15) in the classification of all quadratic systems whose generic complexified periodic orbits define elliptic curves [\textit{S. Gautier, L. Gavrilov} and \textit{I. D. Iliev}, Discrete Contin. Dyn. Syst. 25, No. 2, 511--535 (2009; Zbl 1178.34037)]. By using the method of Abelian integrals, it is proved that the period annulus around the center at the origin has cyclicity two. The same result was obtained earlier (by another method) in the paper of \textit{M. Grau, F. Mañosas} and \textit{J. Villadelprat} [``A Chebyshev criterion for Abelian integrals'', Trans. Am. Math. Soc. 363, No. 1, 109--129 (2011; Zbl 1217.34052)].
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 5968166
zbMATH Keywords: quadratic reversible system
zbMATH Keywords: small perturbation
zbMATH Keywords: limit cycle
zbMATH Keywords: weakened Hilbert 16th problem
zbMATH Keywords: abelian integrals
zbMATH Keywords: period annulus
reviewed by: Iliya D. Iliev
MaRDI profile type: Publication
full work available at URL: s11464-011-0155-4
OpenAlex ID: W2093369129
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DOI: 10.1007/S11464-011-0155-4
description: scientific article; zbMATH DE number 5968166