Local bifurcations of critical periods for cubic Liénard equations with cubic damping
type: Item
instance of: scholarly article
title: Local bifurcations of critical periods for cubic Liénard equations with cubic damping
zbMATH Open document ID: 1163.34349
author: Lan Zou
author: Xingwu Chen
author: Weinian Zhang
published in: Journal of Computational and Applied Mathematics
publication date: 2008-11-18T00:00:00Z
review text: The paper is devoted to the local bifurcation of critical periods near a nondegenerate center \(O(0,0)\) of the Liénard system \[ \dot{x}=y,\quad \dot{y}=-g(x)y - f(x) \] with \(f(x)=a_1x+a_2x^2+a_3x^3\) and \(g(x)=x+b_2x^2+b_3x^3,\) where \(a_1, a_2, a_3, b_2, b_3\in{{\mathbb{R}}}\). The authors first apply the results from \textit{C. Christopher} and \textit{J. Devlin} [J. Differ. Equations 200, No.~1, 1--17 (2004; Zbl 1059.34020)] to give a necessary and sufficient condition for the coefficients under which the cubic Liénard system with cubic damping has a center at \(O\) and finding the set of coefficients in which the center is isochronous. It is proved that at most \(2\) local critical periods can be produced from either a weak center of finite order or the linear isochronous center and that at most one local critical period can be produced from nonlinear isochronous centers.
reviewed by: Alexander Grin
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5368378
zbMATH Keywords: Liénard system
zbMATH Keywords: weak center
zbMATH Keywords: isochronous center
zbMATH Keywords: bifurcation
MaRDI profile type: Publication
full work available at URL: j.cam.2007.11.005
OpenAlex ID: W2107019725
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DOI: 10.1016/J.CAM.2007.11.005
description: scientific article; zbMATH DE number 5368378