The limit cycle of the van der Pol equation is not algebraic
type: Item
instance of: scholarly article
title: The limit cycle of the van der Pol equation is not algebraic
zbMATH Open document ID: 0816.34023
author: Kenzi Odani
published in: Journal of Differential Equations
publication date: 1995-01-30T00:00:00Z
review text: It is shown that no solution curve of the so-called Liénard equation \((y'' + f(y)y' + g(y) = 0)\) lies in an algebraic curve if \(f\) and \(g\) are nonzero polynomials, if \(\deg f \geq \deg g\) and if \(f/g\) is not constant. The proof is based on a curious but elementary lemma that gives a necessary and sufficient condition for a solution curve of a polynomial system of the first order to be a portion of an irreducible algebraic curve. Then a clever degree calculation leads to the result.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 12H20
zbMATH DE Number: 714689
zbMATH Keywords: algebraic solution
zbMATH Keywords: Liénard equation
reviewed by: Gilles Christol
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1008
OpenAlex ID: W2044392679
DOI: 10.1006/JDEQ.1995.1008
Recommended article: On the limit cycle of a generalized van der Pol equation
Recommended article: Q5569705
Recommended article: Bifurcation of discontinuous limit cycles of the Van der Pol equation
Recommended article: Q5409573
Recommended article: Q4346498
Recommended article: Q3712624
Recommended article: Q4885179
Recommended article: Existence of limit cycles for coupled van der Pol equations.
Recommended article: Algebraic limit cycles in polynomial systems of differential equations
Recommended article: Q4251725
description: scientific article; zbMATH DE number 714689