Liénard equations with bounded trajectories in the Liénard plane
type: Item
instance of: scholarly article
title: Liénard equations with bounded trajectories in the Liénard plane
zbMATH Open document ID: 0654.34027
DOI: 10.1016/0022-247X(88)90152-7
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: The simple Liénard equation \(\ddot x+f(x)\dot x+g(x)=0\) is studied in the Liénard plane \((\dot x=y-F(x)\), \(\dot y=-g(x))\). It is assumed that \(F'(x)=f(x)\) and g(x) are continuous functions for all real x. It is shown that there exists some Jordan curve K in the Liénard plane surrounding the point (0,0) such that any trajectory starting in the exterior of K reaches K and eventually stays in the interior of K - provided some extra conditions are fulfilled. The result contains a classical theorem of Dragilev as a special case. If in particular g(x) is odd some further results are derived by separating F(x) in two parts one of which is even.
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 4067370
zbMATH Keywords: bounded global attractors
zbMATH Keywords: Liénard equation
zbMATH Keywords: Jordan curve
author: Kurt Munk Andersen
author: Allan Sandqvist
MaRDI profile type: Publication
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full work available at URL: 0022-247x(88)90152-7
OpenAlex ID: W2105470832
description: scientific article; zbMATH DE number 4067370