Hilbert's 16th problem for quadratic vector fields
type: Item
instance of: scholarly article
title: Hilbert's 16th problem for quadratic vector fields
zbMATH Open document ID: 0802.34028
author: Freddy Dumortier
author: Robert Roussarie
author: Christiane Rousseau
published in: Journal of Differential Equations
publication date: 1994-01-01T00:00:00Z
review text: The second part of Hilbert's 16th problem is to determine the number and relative position of the limit cycles of a polynomial vector field in the plane. This problem remains open even for the case of a vector field whose components are quadratic polynomials. Perhaps the most important and deepest result in the direction of Hilbert's problem is the recent solution of the Dulac problem, solved independently by J. Ecalle and Yu. Il'yashenko. In particular, this work has resulted in the following theorem: A polynomial vector field in the plane has at most a finite number of limit cycles. For a history and a proof of this theorem, see [\textit{Yu. Il'yashenko}, Finiteness theorems for limit cycles, Transl. of Mathematical Monographs, Vol. 94, Am. Math. Soc. (1991; Zbl 0743.34036)]. In view of the solution of Dulac's problem, the unresolved portion of Hilbert's problem is usually stated as follows: For each fixed integer \(N\geq 2\), determine the maximum number of limit cycles for a polynomial vector field of degree \(N\). However, as an intermediate result, it also remains to show that, for each fixed degree, there exists a uniform bound for the number of limit cycles. This is the problem addressed in the paper under review for the case of quadratic vector fields. The main idea underlying the paper is that the existence of a uniform bound on the number of limit cycles can be proved from local finiteness results. In particular, it is sufficient to determine the ``cyclicity'' of certain graphics, that is, certain invariant sets formed by rest points and connecting orbits. A familiar example of the cyclicity method is provided by Bautin's theorem. It addresses the case, where the graphic is a single point, namely, a weak focus or a center of a quadratic system. To state this result, let \(X\) denote a quadratic system with a weak focus or a center. The cyclicity of this rest point in the space of quadratic vector fields is the maximum number of limit cycles that can appea…
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 599886
zbMATH Keywords: cyclicity
zbMATH Keywords: second part of Hilbert's 16th problem
zbMATH Keywords: number and relative position of the limit cycles
zbMATH Keywords: polynomial vector field in the plane
zbMATH Keywords: Dulac problem
zbMATH Keywords: quadratic vector fields
zbMATH Keywords: Bautin's theorem
zbMATH Keywords: weak focus
zbMATH Keywords: center
zbMATH Keywords: bifurcation
reviewed by: C. Chicone
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1061
OpenAlex ID: W2054609165
DOI: 10.1006/JDEQ.1994.1061
description: scientific article; zbMATH DE number 599886