Simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli

Leonardo P.C. da Cruz, Joan Torregrosa

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4 Citations (Scopus)


© 2018 Elsevier Inc. We study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations. The cubic planar system (x′,y′)=(−y((x−1)2+y2),x((x−1)2+y2)) has simultaneously a center at the origin and at infinity. We study, up to first order averaging analysis, the bifurcation of periodic orbits from the two period annuli, first separately and second simultaneously. This problem is a generalization of [24] to the piecewise systems class. When the polynomial perturbation has degree n, we prove that the inner and outer Abelian integrals are rational functions and we provide an upper bound for the number of zeros. When the perturbation is cubic, the same degree as the unperturbed vector field, the maximum number of limit cycles, up to first order perturbation, from the inner and outer annuli is 9 and 8, respectively. When the simultaneous bifurcation problem is considered, 12 limit cycles exist. These limit cycles appear in three types of configurations: (9,3), (6,6) and (4,8). In the non-piecewise scenario, only 5 limit cycles were found.
Original languageEnglish
Pages (from-to)248-272
JournalJournal of Mathematical Analysis and Applications
Issue number1
Publication statusPublished - 1 May 2018


  • Limit cycles
  • Piecewise vector field
  • Simultaneous bifurcation
  • Zeros of Abelian integrals


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