Explicit non-algebraic limit cycles for polynomial systems

A. Gasull, H. Giacomini, J. Torregrosa

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40 Citations (Scopus)


We consider a system of the form over(x, ̇) = Pn (x, y) + xRm (x, y), over(y, ̇) = Qn (x, y) + yRm (x, y), where Pn (x, y), Qn (x, y) and Rm (x, y) are homogeneous polynomials of degrees n, n and m, respectively, with n ≤ m. We prove that this system has at most one limit cycle and that when it exists it can be explicitly found and given by quadratures. Then we study a particular case, with n = 3 and m = 4. We prove that this quintic polynomial system has an explicit limit cycle which is not algebraic. To our knowledge, there are no such type of examples in the literature. The method that we introduce to prove that this limit cycle is not algebraic can be also used to detect algebraic solutions for other families of polynomial vector fields or for probing the absence of such type of solutions. © 2006 Elsevier B.V. All rights reserved.
Original languageEnglish
Pages (from-to)448-457
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - 1 Mar 2007


  • Limit cycle
  • Non-algebraic solution
  • Polynomial planar system


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