Bifurcation of limit cycles in the neighborhood of generic hyperbolic polycycles with three vertices
type: Item
instance of: scholarly article
title: Bifurcation of limit cycles in the neighborhood of generic hyperbolic polycycles with three vertices
zbMATH Open document ID: 0805.58053
author: Abderaouf Mourtada
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1995-02-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73335
review text: The author, a brilliant student of R. Roussarie (Dijon) works on the 16th Hilbert problem, with the aim of obtaining explicit bounds on the number of limit cycles bifurcating from some type of an elementary polycycle [the author, Lect. Notes Math. 1455, 272-314 (1990; Zbl 0719.58031) and Ann. Inst. Fourier 41, No. 3, 719-753 (1991; Zbl 0725.58032)]. In the present paper a very important result is obtained. It is shown that generic hyperbolic polycycles with three vertices are of cyclicity at most three in \(C^ \infty\) families of vector fields on the plane. A complete topological classification is given. As a counterexample, a class of degenerate hyperbolic polycycles of cyclicity at least four is presented. The paper is virtually self-contained and well written. The source of inspiration was certainly the work of \textit{N. N. Bautin} [Mat. Sb., n. Ser. 30(72), 181-196 (1952; Zbl 0046.094)] and that of \textit{R. Roussarie} [Bol. Soc. Bras. Mat. 17, No. 1, 67-101 (1986; Zbl 0628.34032)], but the proofs go far beyond the sources, with new ideas and new methods, which are not all easy or obvious. This work is worth reading and learning from.
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 635811
zbMATH Keywords: bifurcation
zbMATH Keywords: 16th Hilbert problem
zbMATH Keywords: limit cycles
zbMATH Keywords: hyperbolic polycycles
reviewed by: Zofia Denkowska
MaRDI profile type: Publication
OpenAlex ID: W2327291577
cites work: Q5832271
cites work: Q4722875
cites work: Keeping track of limit cycles
cites work: Generalized Hopf Bifurcation and Its Dual Generalized Homoclinic Bifurcation
cites work: Q5751632
cites work: Cyclicité finie des polycycles hyperboliques de champs de vecteurs du plan. Algorithme de finitude. (Finite cyclicity of hyperbolic polycycles of plane vector fields. Finiteness algorithm)
cites work: Q3831770
cites work: On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields
DOI: 10.5802/AFST.780
description: scientific article; zbMATH DE number 635811