Limit cycles bifurcating from a degenerate center

Jaume Llibre, Chara Pantazi

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6 Citations (Scopus)


© 2015 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved. We study the maximum number of limit cycles that can bifurcate from a degenerate center of a cubic homogeneous polynomial differential system. Using the averaging method of second order and perturbing inside the class of all cubic polynomial differential systems we prove that at most three limit cycles can bifurcate from the degenerate center. As far as we know this is the first time that a complete study up to second order in the small parameter of the perturbation is done for studying the limit cycles which bifurcate from the periodic orbits surrounding a degenerate center (a center whose linear part is identically zero) having neither a Hamiltonian first integral nor a rational one. This study needs many computations, which have been verified with the help of the algebraic manipulator Maple.
Original languageEnglish
Pages (from-to)1-11
JournalMathematics and Computers in Simulation
Publication statusPublished - 17 Jul 2015


  • Averaging theory
  • Centers
  • Limit cycles
  • Polynomial differential systems


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