Liouvillian Integrability Versus Darboux Polynomials

Jaume Llibre; Claudia Valls; Xiang Zhang

doi:https://doi.org/10.1007/s12346-016-0212-1

Liouvillian Integrability Versus Darboux Polynomials

Jaume Llibre, Claudia Valls, Xiang Zhang

Department of Mathematics

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

3 Citations (Scopus)

Abstract

© 2016, Springer International Publishing. In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Liénard polynomial differential system x˙=y,y˙=-f(x)y-g(x) with deg f> deg g is not Liouvillian integrable.

Original language	English
Pages (from-to)	503-515
Journal	Qualitative Theory of Dynamical Systems
Volume	15
Issue number	2
DOIs	https://doi.org/10.1007/s12346-016-0212-1
Publication status	Published - 1 Oct 2016

Keywords

Darboux Jacobian multiplier
Darboux polynomial
Liouvillian integrability
Polynomial differential system

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https://doi.org/10.1007/s12346-016-0212-1

https://ddd.uab.cat/record/169448

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title = "Liouvillian Integrability Versus Darboux Polynomials",

abstract = "{\textcopyright} 2016, Springer International Publishing. In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Li{\'e}nard polynomial differential system x˙=y,y˙=-f(x)y-g(x) with deg f> deg g is not Liouvillian integrable.",

keywords = "Darboux Jacobian multiplier, Darboux polynomial, Liouvillian integrability, Polynomial differential system",

author = "Jaume Llibre and Claudia Valls and Xiang Zhang",

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T1 - Liouvillian Integrability Versus Darboux Polynomials

AU - Llibre, Jaume

AU - Valls, Claudia

AU - Zhang, Xiang

PY - 2016/10/1

Y1 - 2016/10/1

N2 - © 2016, Springer International Publishing. In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Liénard polynomial differential system x˙=y,y˙=-f(x)y-g(x) with deg f> deg g is not Liouvillian integrable.

AB - © 2016, Springer International Publishing. In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Liénard polynomial differential system x˙=y,y˙=-f(x)y-g(x) with deg f> deg g is not Liouvillian integrable.

KW - Darboux Jacobian multiplier

KW - Darboux polynomial

KW - Liouvillian integrability

KW - Polynomial differential system

U2 - https://doi.org/10.1007/s12346-016-0212-1

DO - https://doi.org/10.1007/s12346-016-0212-1

M3 - Article

SN - 1575-5460

VL - 15

SP - 503

EP - 515

JO - Qualitative Theory of Dynamical Systems

JF - Qualitative Theory of Dynamical Systems

IS - 2

ER -

Liouvillian Integrability Versus Darboux Polynomials

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Keywords

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