On the critical periods of Liénard systems with cubic restoring forces
type: Item
instance of: scholarly article
title: On the critical periods of Liénard systems with cubic restoring forces
zbMATH Open document ID: 1126.34328
DOI: 10.1155/S0161171204402245
author: Zhengdong Du
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2005-05-25T00:00:00Z
full work available at URL: https://eudml.org/doc/53591
review text: Consider the Liénard system \[ \dot x= -y+ F(x),\qquad\dot y= g(x), \] where \(F\) and \(g\) are polynomials with \(\deg(g)\leq 3\), \(g(0)= 0\), \(g'(0)= 1\), \(F(0)= F'(0)= 0\) such that the origin is a non-degenerate center. The author introduces the notion ``weak center of type I or type II''. The period function of a center associates to each point (\(x= r\), \(y= 0\)) near the origin the primitive period \(T(r)\) of the closed orbit passing the point \((r,0)\). A critical period corresponds to a solution of \(T'(r)= 0\). The author estimates the number of local critical periods which can bifurcate. He shows that for a weak center of type I, at most \([{\deg(F)\over 2}]-1\) local critical periods can bifurcate, while for a weak center of type II at least \([{deg(F)\over 4}]\) can bifurcate.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G10
zbMATH DE Number: 2171777
MaRDI profile type: Publication
OpenAlex ID: W1975427660
description: scientific article; zbMATH DE number 2171777