Generalized van der Pol equation and Hilbert's 16th problem
type: Item
instance of: scholarly article
zbMATH Open document ID: 1309.34046
author: Xenakis Ioakim
publication date: 2014-08-25T00:00:00Z
full work available at URL: abstr.html
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 6333961
zbMATH Keywords: generalized van der Pol equation
zbMATH Keywords: Hilbert's 16th problem
zbMATH Keywords: limit cycle
zbMATH Keywords: existence
zbMATH Keywords: sinusoidal-type number
zbMATH Keywords: sinusoidal-type set
zbMATH Keywords: dependent radius
zbMATH Keywords: \(\Lambda\)-point
MaRDI profile type: Publication
title: Generalized van der Pol equation and Hilbert's 16th problem
published in: Electronic Journal of Differential Equations (EJDE)
review text: The problem under consideration is an upper bound for the number of limit cycles bifurcating from the linear center in the generalized Van der Pol system NEWLINE\[NEWLINE \dot{x}=y, \quad \dot{y}=-x +\varepsilon f(y)(1-x^{2}),NEWLINE\]NEWLINE when the parameter \(\varepsilon\) crosses zero value. The author proves that \(n+1\) is an upper bound for the number of limit cycles for the case that \(f\) is an odd polynomial \(f\) of degree \(2n+1\). For the proof the Poincaré method and some results of Iliev are applied. It is shown how to construct a polynomial \(f\) such that the upper bound is attained. Some examples illustrating the presented theory are included too.
reviewed by: Alexander Grin
description: scientific article; zbMATH DE number 6333961