On saddle-loop bifurcations of limit cycles in perturbations of quadratic Hamiltonian systems
type: Item
instance of: scholarly article
title: On saddle-loop bifurcations of limit cycles in perturbations of quadratic Hamiltonian systems
zbMATH Open document ID: 0808.34041
author: Emil Horozov
author: Iliya D. Iliev
published in: Journal of Differential Equations
publication date: 1994-11-28T00:00:00Z
review text: This paper studies the bifurcations (of saddle-loop connections) that arise from a perturbation of quadratic Hamiltonian vector fields on the plane. The problem is closely related to the second part of Hilbert's 16th problem concerning the number of limit cycles of polynomial vector fields. Guckenheimer et al. have conjectured that the number of limit cycles of vector fields of the form \(X_ \varepsilon (x,y) = (H_ y + \varepsilon f,-H_ x + \varepsilon g)\) (where \(\varepsilon\) is small and \(X_ 0\) is a quadratic Hamiltonian vector field) does not exceed two and gave numerical evidence for that. In this present paper this conjecture is proved.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 665723
zbMATH Keywords: bifurcations
zbMATH Keywords: quadratic Hamiltonian vector fields on the plane
zbMATH Keywords: Hilbert's 16th problem
zbMATH Keywords: number of limit cycles
zbMATH Keywords: polynomial vector fields
reviewed by: Marco A. Teixeira
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1115
OpenAlex ID: W1997634909
DOI: 10.1006/JDEQ.1994.1115
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description: scientific article; zbMATH DE number 665723