Perturbation from an elliptic Hamiltonian of degree four. IV: Figure eight-loop.
type: Item
instance of: scholarly article
title: Perturbation from an elliptic Hamiltonian of degree four. IV: Figure eight-loop.
zbMATH Open document ID: 1057.34015
DOI: 10.1016/S0022-0396(02)00111-0
author: Freddy Dumortier
author: Chengzhi Li
published in: Journal of Differential Equations
publication date: 2003-04-28T00:00:00Z
review text: From the paper's abstract: ``The paper deals with Liénard equations of the form \(\dot x=y\), \(\dot y = P(x)+yQ(x)\) with \(P\) and \(Q\) polynomials of degree, 3 and 2, respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree four, exhibiting a figure eight-loop. It is proved that the least upper bound of the number of zeroes of the related elliptic integral is five, and this is a sharp bound, multiplicity taken into account. Moreover, if restricting to the level curves `inside' a saddle loop, or `outside' the figure eight-loop, the sharp upper bound is two or four, respectively; also, the multiplicity of the zeroes is at most four.'' The authors also show the existence of a cubic Liénard system with five limit cycles. This paper is the fourth one of four papers devoted to the study of perturbations of planar Hamiltonian vector fields. In the three previous papers [Part III, see ibid. 188, No. 2, 473-511 (2003; Zbl 1056.34044)], the authors studied the saddle loop, the two saddle cycle, the cuspidal loop and the global centre cases.
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 1902859
zbMATH Keywords: eight-loop
zbMATH Keywords: limit cycle
zbMATH Keywords: Liénard
zbMATH Keywords: Hamiltonian
reviewed by: Marco Sabatini
MaRDI profile type: Publication
cites work: Q4305607
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cites work: Perturbations from an elliptic Hamiltonian of degree four. II: Cuspidal loop
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description: scientific article; zbMATH DE number 1902859