On the number of limit cycles for perturbed pendulum equations

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© 2016 The Authors. We consider perturbed pendulum-like equations on the cylinder of the form where Qn,s are trigonometric polynomials of degree n, and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case ε=0 in terms of m and n. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. Some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be Chebyshev systems.
Original languageEnglish
Pages (from-to)2141-2167
JournalJournal of Differential Equations
Issue number3
Publication statusPublished - 5 Aug 2016


  • Abelian integrals
  • Infinitesimal Sixteenth Hilbert problem
  • Limit cycles
  • Perturbed pendulum equation
  • Primary
  • Secondary


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