The Hilbert number of a class of differential equations

Jaume Llibre, Ammar Makhlouf

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© 2015, Journal of Applied Analysis and Computation. All Rights Reserved. The notion of Hilbert number from polynomial differential systems in the plane of degree n can be extended to the differential equations of the form (formula presented) defined in the region of the cylinder (θ; r) ∈ θS<sup>1</sup>× ℝ where the denominator of (*) does not vanish. Here a; a<inf>0</inf>; a<inf>1</inf>;...; a<inf>n</inf> are analytic 2π–periodic functions, and the Hilbert number ℍ (n) is the supremum of the number of limit cycles that any differential equation (*) on the cylinder of degree n in the variable r can have. We prove that ℍ (n) = ∞ for all n ≥ 1.
Original languageEnglish
Pages (from-to)141-145
JournalJournal of Applied Analysis and Computation
Issue number1
Publication statusPublished - 1 Jan 2015


  • Averaging theory
  • Hilbert number
  • Periodic orbit
  • Trigonometric polynomial


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