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Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

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Abstract

This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems \(\mathop x\limits^. \) = λ x−y +P n+1(x, y)+xF 2n (x, y), \(\mathop y\limits^. \) = x+λ y +Q n+1(x, y)+yF 2n (x, y), where P i (x, y), Q i (x, y) and F i (x, y) are homogeneous polynomials of degree i. Within this class, we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

    JinLong Cao & Xiang Zhang

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

    Jaume Llibre

  3. Ministry of Education Key Lab of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai, 200240, China

    Xiang Zhang

Authors
  1. JinLong Cao
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  2. Jaume Llibre
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  3. Xiang Zhang
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Correspondence to Xiang Zhang.

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Cao, J., Llibre, J. & Zhang, X. Darboux integrability and algebraic limit cycles for a class of polynomial differential systems. Sci. China Math. 57, 775–794 (2014). https://doi.org/10.1007/s11425-014-4772-8

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  • DOI: https://doi.org/10.1007/s11425-014-4772-8

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