Limit cycles in quadratic systems with a first-order weak focus
type: Item
instance of: scholarly article
title: Limit cycles in quadratic systems with a first-order weak focus
zbMATH Open document ID: 1273.34041
DOI: 10.1134/S1064562410060220
published in: Doklady Mathematics
publication date: 2011-06-08T00:00:00Z
review text: Consider the quadratic polynomial system \[ \begin{aligned} \dot x &= x^2+ xy+ y,\\ \dot y &= ax^2+ bxy+ cy^2- {1\over\varepsilon} x\end{aligned}\tag{\(*\)} \] under the conditions \[ c\in\Biggl({1\over 3}, 1\Biggr),\;c\neq{1\over 2},\;bc< 1,\;b> a+ c,\;2c< b+1, \] \[ 4a(c-1)> (b-1)^2,\quad 0<\varepsilon\ll 1. \] The author proves that under these assumptions system \((*)\) has three limit cycles, one limit cycle on the left of the straight line \(x=-1\), and two cycles on the right of this line. The proof is based on the reduction of \((*)\) to a Liénard system and studying the corresponding phase portrait.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 5905507
author: G. A. Leonov
MaRDI profile type: Publication
cites work: Synthesis of two-dimensional quadratic systems with a limit cycle satisfying prescribed initial conditions
cites work: HILBERT'S 16TH PROBLEM FOR QUADRATIC SYSTEMS: NEW METHODS BASED ON A TRANSFORMATION TO THE LIENARD EQUATION
cites work: Necessary and sufficient conditions for the boundedness of solutions to two-dimensional quadratic systems in a positively invariant half-plane
full work available at URL: s1064562410060220
OpenAlex ID: W2067993798
description: scientific article; zbMATH DE number 5905507