Darboux integrability and the inverse integrating factor

Javier Chavarriga, Hector Giacomini, Jaume Giné, Jaume Llibre

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77 Citations (Scopus)


We mainly study polynomial differential systems of the form dx/dt = P(x, y), dy/dt = Q(x, y), where P and Q are complex polynomials in the dependent complex variables x and y, and the independent variable t is either real or complex. We assume that the polynomials P and Q are relatively prime and that the differential system has a Darboux first integral of the form where the polynomials fi and gj are irreducible, the polynomials gj and hj are coprime, and the λi and μj are complex numbers, when i = 1, ...,p and j = 1, ...,q. A figure is presented. Prelle and Singer proved that these systems have a rational integrating factor. We improve this result as follows. Assume that H is a rational function which is not polynomial. Following to Poincaré we define the critical remarkable values of H. Then, we prove that the system has a polynomial inverse integrating factor if and only if H has at most two critical remarkable values. Under some assumptions over the Darboux first integral H we show, first that the system has a polynomial inverse integrating factor; and secondly that if the degree of the system is m, the homogeneous part of highest degree of H is a multi-valued function, and the functions exp(hj/gj) are exponential factors for j = 1, ...,q, then the system has a polynomial inverse integrating factor of degree m + 1. We also present versions of these results for real polynomial differential systems. Finally, we apply these results to real polynomial differential systems having a Darboux first integral and limit cycles or foci. © 2003 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)116-139
JournalJournal of Differential Equations
Publication statusPublished - 10 Oct 2003


  • Darboux integrability
  • Inverse integrating factor
  • Polynomial first integral
  • Rational first integral


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