Upper bounds for the number of zeroes for some Abelian integrals

Armengol Gasull, J. Tomás Lázaro, Joan Torregrosa

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28 Citations (Scopus)


Consider the vector field x′=-yG(x,y), y′= xG(x,y), where the set of critical points G(x,y)=0 is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4 we recover or improve some results obtained in several previous works. © 2012 Elsevier Ltd. All rights reserved.
Original languageEnglish
Pages (from-to)5169-5179
JournalNonlinear Analysis, Theory, Methods and Applications
Issue number13
Publication statusPublished - 1 Sept 2012


  • Abelian integrals
  • Chebyshev system
  • Limit cycles
  • Number of zeroes of real functions
  • Weak 16th Hilbert's Problem


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