Seeking Darboux Polynomials

Antoni Ferragut; Armengol Gasull

doi:https://doi.org/10.1007/s10440-014-9974-0

Seeking Darboux Polynomials

Antoni Ferragut, Armengol Gasull

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

7 Citations (Scopus)

Abstract

© 2014, Springer Science+Business Media Dordrecht. We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in $\mathbb {R}^{n}$. We apply these techniques to some well-known systems when n=2,3,4. We also propose a general method for computing Darboux polynomials in the plane. As an application we prove that a family of potential systems, that includes the van der Pol one, has no Darboux polynomials, giving in particular a new simple proof that the van der Pol limit cycle is not algebraic.

Original language	English
Pages (from-to)	167-186
Journal	Acta Applicandae Mathematicae
Volume	139
Issue number	1
DOIs	https://doi.org/10.1007/s10440-014-9974-0
Publication status	Published - 29 Oct 2015

Keywords

Birational map
Cofactor
Darboux polynomial
Non-algebraic limit cycle
Planar polynomial differential system

Access to Document

https://doi.org/10.1007/s10440-014-9974-0

ddd.uab.cat Link

Fingerprint Dive into the research topics of 'Seeking Darboux Polynomials'. Together they form a unique fingerprint.

Cite this

@article{d1809f351dac4a5aa97b10c8970b5340,

title = "Seeking Darboux Polynomials",

abstract = "{\textcopyright} 2014, Springer Science+Business Media Dordrecht. We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in $\mathbb {R}^{n}$. We apply these techniques to some well-known systems when n=2,3,4. We also propose a general method for computing Darboux polynomials in the plane. As an application we prove that a family of potential systems, that includes the van der Pol one, has no Darboux polynomials, giving in particular a new simple proof that the van der Pol limit cycle is not algebraic.",

keywords = "Birational map, Cofactor, Darboux polynomial, Non-algebraic limit cycle, Planar polynomial differential system",

author = "Antoni Ferragut and Armengol Gasull",

year = "2015",

month = oct,

day = "29",

doi = "https://doi.org/10.1007/s10440-014-9974-0",

language = "English",

volume = "139",

pages = "167--186",

journal = "Acta Applicandae Mathematicae",

issn = "0167-8019",

number = "1",

}

TY - JOUR

T1 - Seeking Darboux Polynomials

AU - Ferragut, Antoni

AU - Gasull, Armengol

PY - 2015/10/29

Y1 - 2015/10/29

N2 - © 2014, Springer Science+Business Media Dordrecht. We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in $\mathbb {R}^{n}$. We apply these techniques to some well-known systems when n=2,3,4. We also propose a general method for computing Darboux polynomials in the plane. As an application we prove that a family of potential systems, that includes the van der Pol one, has no Darboux polynomials, giving in particular a new simple proof that the van der Pol limit cycle is not algebraic.

AB - © 2014, Springer Science+Business Media Dordrecht. We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in $\mathbb {R}^{n}$. We apply these techniques to some well-known systems when n=2,3,4. We also propose a general method for computing Darboux polynomials in the plane. As an application we prove that a family of potential systems, that includes the van der Pol one, has no Darboux polynomials, giving in particular a new simple proof that the van der Pol limit cycle is not algebraic.

KW - Birational map

KW - Cofactor

KW - Darboux polynomial

KW - Non-algebraic limit cycle

KW - Planar polynomial differential system

UR - https://ddd.uab.cat/record/145343

U2 - https://doi.org/10.1007/s10440-014-9974-0

DO - https://doi.org/10.1007/s10440-014-9974-0

M3 - Article

VL - 139

SP - 167

EP - 186

JO - Acta Applicandae Mathematicae

JF - Acta Applicandae Mathematicae

SN - 0167-8019

IS - 1

ER -