On the center of the Liénard equation
type: Item
instance of: scholarly article
title: On the center of the Liénard equation
zbMATH Open document ID: 0781.34022
author: Jizhou Zhang
author: Shuxiang Yu
published in: Journal of Differential Equations
publication date: 1993-10-18T00:00:00Z
review text: The authors study the existence of closed orbits (in the real \(xy\)-plane) of the system \(dx/dt=y-F(x)\), \(dy/dt=x\), both locally near the origin (assumed to be the unique critical point) and globally throughout the plane. They first give two examples which show that the criteria derived by \textit{P. J. Ponzo} and \textit{N. Wax} [J. Differ. Equations 10, 262-269 (1971; Zbl 0222.34045)] for the origin to be a local or alternatively a global center are not sufficient. They then give sufficient conditions (on \(F'(x)\) and \(g(x)\)) under which the origin is a local or global center. The proof relies on qualitative results of dynamical systems to be found in two monographs: \textit{G. Sansone} and \textit{R. Conti} [Nonlinear differential equations, New York (1964; Zbl 0128.084)] and \textit{A. A. Andronov}, \textit{E. A. Leontovich}, \textit{I. I. Gordon} and \textit{A. G. Maier} [Qualitative theory of second-order dynamic systems, New York (1973; Zbl 0282.34022)].
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 425434
zbMATH Keywords: Liénard equation
zbMATH Keywords: closed orbits
zbMATH Keywords: global center
reviewed by: Nalini Joshi
MaRDI profile type: Publication
full work available at URL: jdeq.1993.1021
OpenAlex ID: W2000161448
DOI: 10.1006/JDEQ.1993.1021
description: scientific article; zbMATH DE number 425434