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Seeking Darboux Polynomials

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Abstract

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.

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Acknowledgements

The authors are partially supported by Spanish Government MTM2013-40998-P grant. The second author is also partially supported by Generalitat de Catalunya Government 2014SGR568 grant.

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Authors and Affiliations

  1. Institut de Matemàtiques i Aplicacions de Castelló (IMAC) and Departament de Matemàtiques, Universitat Jaume I, Edifici TI (ESTEC), Av. de Vicent Sos Baynat, s/n, Campus del Riu Sec, 12071, Castelló de la Plana, Spain

    Antoni Ferragut

  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193, Bellaterra, Barcelona, Catalonia, Spain

    Armengol Gasull

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  1. Antoni Ferragut
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  2. Armengol Gasull
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Correspondence to Antoni Ferragut.

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Ferragut, A., Gasull, A. Seeking Darboux Polynomials. Acta Appl Math 139, 167–186 (2015). https://doi.org/10.1007/s10440-014-9974-0

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