A new criterion for controlling the number of limit cycles of some generalized Liénard equations
type: Item
instance of: scholarly article
title: A new criterion for controlling the number of limit cycles of some generalized Liénard equations
zbMATH Open document ID: 1032.34028
author: Armengol Gasull
author: Hector Giacomini
published in: Journal of Differential Equations
publication date: 2003-03-18T00:00:00Z
review text: The authors consider the planar differential system \[ \dot x=p(x) +q(x)y, \qquad \dot y=r(x)+s(x)y+ t(x)y^*y, \tag{1} \] which includes the Liénard system \[ \dot x= y-f(x), \qquad \dot y=-g(x). \tag{2} \] The number of limit cycles of (1) is studied. The authors show that the problem can be reduced to the condition that a function, \(M(x)\), does not change sign. An application of the main result gives a sharp upper bound for the number of limit cycles of some systems (2), especially, yields exactly 3 limit cycles for the system \[ \dot x= y-x(x^*x-2)(x^*x-1) (x^*x-1/4), \qquad \dot y=-x. \tag{3} \] The reviewer points out that applying a previous result on the odd polynomial Liénard system to (3) gives immediately that (3) has at most 3 limit cycles [\textit{Y. Chen} and \textit{X. Chen}, J. Math. Res. Expo. 22, 368-370 (2002; Zbl 1074.34032)].
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 1884194
zbMATH Keywords: Liénard equations
zbMATH Keywords: limit cycles
reviewed by: Jinghuang Tian
MaRDI profile type: Publication
full work available at URL: jdeq.2002.4172
OpenAlex ID: W2047605647
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DOI: 10.1006/JDEQ.2002.4172
description: scientific article; zbMATH DE number 1884194