On the 16th Hilbert problem for algebraic limit cycles

Jaume Llibre, Rafael Ramírez, Natalia Sadovskaia

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24 Citations (Scopus)


For a polynomial planar vector field of degree n ≥ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (n - 1) (n - 2) / 2 when n is even, and (n - 1) (n - 2) / 2 when n is odd. Furthermore, these upper bounds are reached. © 2009 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)1401-1409
JournalJournal of Differential Equations
Publication statusPublished - 15 Mar 2010


  • 16th Hilbert problem
  • Algebraic limit cycles
  • Limit cycles
  • Polynomial vector fields


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