On the number of limit cycles of planar quadratic vector fields
type: Item
instance of: scholarly article
title: On the number of limit cycles of planar quadratic vector fields
zbMATH Open document ID: 1192.34042
DOI: 10.1134/S1064562409050251
author: A. Yu. Fishkin
published in: Doklady Mathematics
publication date: 2010-01-12T00:00:00Z
review text: The paper is devoted to restricted Hilbert's 16th problem for quadratic vector fields which can be transformed to the Kapteyn normal form \[ \dot{x}=\lambda_1x-y-\lambda_3x^2+(2\lambda_2+\lambda_5)xy+\lambda_6y^2,\quad \dot{y}=x+\lambda_1y+\lambda_2x^2+(2\lambda_3+\lambda_4)xy-\lambda_2y^2, \] where \(\lambda_1\in\mathbb{R},\) \(\bar{\lambda}=(\lambda_2,\dots,\lambda_6)\in\mathbb{S}^5.\) The author considers limit cycles surrounding the origin which do not pass the \(\delta\)-neighborhoods of all singular points and that are contained in a disc \(\sqrt{x^2+y^2}\leq1/\delta,\) \(\delta>0\) is arbitrary. The main result estimates the Hilbert number \(H(\delta,\lambda)\) under certain conditions uniformly in \(\lambda\).
reviewed by: Alexander Grin
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5656131
zbMATH Keywords: quadratic vector field
zbMATH Keywords: restricted Hilbert's 16th problem
zbMATH Keywords: limit cycle
zbMATH Keywords: Kapteyn normal form
zbMATH Keywords: singular point
MaRDI profile type: Publication
cites work: Centennial History of Hilbert's 16th Problem
cites work: Hilbert's 16th problem for quadratic vector fields
cites work: A survey of quadratic systems
cites work: Bifurcation of planar vector fields and Hilbert's sixteenth problem
cites work: Q5720346
cites work: Counting real zeros of analytic functions satisfying linear ordinary differential equations
full work available at URL: s1064562409050251
OpenAlex ID: W1998110975
description: scientific article; zbMATH DE number 5656131