Liouvillian Integrability Versus Darboux Polynomials

• Jaume Llibre ; Claudia Valls ^[2] ; Xiang Zhang ^[1]
  1. [1] Shanghai Jiao Tong University

     Shanghai Jiao Tong University

     China

     
  2. [2] Universidade Técnica de Lisboa (Portugal)
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 15, Nº 2, 2016, págs. 503-515
• Idioma: inglés
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• Resumen

  • 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.