Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones

Jaume Llibre, Douglas D. Novaes, Marco A. Teixeira

Research output: Contribution to journalArticleResearchpeer-review

29 Citations (Scopus)

Abstract

© 2015 World Scientific Publishing Company. We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N > 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N > 3.
Original languageEnglish
Article number1550144
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume25
Issue number11
DOIs
Publication statusPublished - 1 Oct 2015

Keywords

  • Discontinuous differential system
  • limit cycle
  • piecewise linear differential system

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