The 16th Hilbert problem restricted to circular algebraic limit cycles

Jaume Llibre, Rafael Ramírez, Valentín Ramírez, Natalia Sadovskaia

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13 Citations (Scopus)

Abstract

© 2015 Elsevier Inc. We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.
Original languageEnglish
Pages (from-to)5726-5760
JournalJournal of Differential Equations
Volume260
Issue number7
DOIs
Publication statusPublished - 5 Apr 2016

Keywords

  • 16th Hilbert's problem
  • Algebraic limit circles
  • Darboux integrability
  • Invariant algebraic circles
  • Planar polynomial differential system
  • Polynomial vector fields

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