Piecewise linear perturbations of a linear center

Claudio Buzzi, Claudio Pessoa, Joan Torregrosa

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Abstract

This paper is mainly devoted to the study of the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. We consider the case when the two zones are separated by a straight line Σ and the singular point of the unperturbed system is in Σ. It is proved that the maximum number of limit cycles that can appear up to a seventh order perturbation is three. Moreover this upper bound is reached. This result confirms that these systems have more limit cycles than it was expected. Finally, center and isochronicity problems are also studied in systems which include a first order perturbation. For the latter systems it is also proved that, when the period function, defined in the period annulus of the center, is not monotone, then it has at most one critical period. Moreover this upper bound is also reached.
Original languageEnglish
Pages (from-to)3915-3936
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume33
Issue number9
DOIs
Publication statusPublished - 1 Sep 2013

Keywords

  • Centers
  • Isochronicity
  • Limit cycle
  • Non-smooth differential system
  • Piecewise linear differential system

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    Buzzi, C., Pessoa, C., & Torregrosa, J. (2013). Piecewise linear perturbations of a linear center. Discrete and Continuous Dynamical Systems- Series A, 33(9), 3915-3936. https://doi.org/10.3934/dcds.2013.33.3915