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Liouvillian Integrability Versus Darboux Polynomials

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Abstract

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 \(\dot{x}=y,\, \dot{y}=-f(x)y-g(x)\) with \(\deg f>\deg g\) is not Liouvillian integrable.

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Acknowledgments

J. Llibre is partially supported by a MINECO Grant MTM2013-40998-P, an AGAUR Grant Number 2014SGR-568, and the Grants FP7-PEOPLE-2012-IRSES 318999 and 316338. C. Valls is supported by Portuguese national funds through FCT—Fundação para a Ciência e a Tecnologia: project PEst-OE/EEI/LA0009/2013 (CAMGSD). X. Zhang is partially supported by NNSF of China Grant Numbers 11271252 and 11671254, by RFDP of Higher Education of China Grant Number 20110073110054, and by FP7-PEOPLE-2012-IRSES-316338 of Europe. This work was done during a visit of the C. Valls and X. Zhang to the Universitat Autònoma de Barcelona. We appreciate their support and hospitality. We thank to the reviewer his/her comments which help us to improve the presentation of this paper.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Claudia Valls

  3. Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Xiang Zhang

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  1. Jaume Llibre
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  2. Claudia Valls
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  3. Xiang Zhang
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Correspondence to Jaume Llibre.

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Llibre, J., Valls, C. & Zhang, X. Liouvillian Integrability Versus Darboux Polynomials. Qual. Theory Dyn. Syst. 15, 503–515 (2016). https://doi.org/10.1007/s12346-016-0212-1

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