On the limit cycles bifurcating from an ellipse of a quadratic center

Jaume Llibre, Dana Schlomiuk

Research output: Contribution to journalArticleResearchpeer-review

4 Citations (Scopus)

Abstract

It is well known that invariant algebraic curves of polynomial differential systems play an important role in questions regarding integrability of these systems. But do they also have a role in relation to limit cycles? In this article we show that not only they do have a role in the production of limit cycles in polynomial perturbations of such systems but that algebraic invariant curves can even generate algebraic limit cycles in such perturbations. We prove that when we perturb any quadratic system with an invariant ellipse surrounding a center (quadratic systems with center always have invariant algebraic curves and some of them have invariant ellipses) within the class of quadratic differential systems, there is at least one 1-parameter family of such systems having a limit cycle bifurcating from the ellipse. Therefore the cyclicity of the period annulus of such systems is at least one.
Original languageEnglish
Pages (from-to)1091-1102
JournalDiscrete and Continuous Dynamical Systems
Volume35
Issue number3
DOIs
Publication statusPublished - 1 Jan 2015

Keywords

  • Bifurcation from center
  • Cyclicity of the period annulus
  • Inverse integrating factor
  • Limit cycle
  • Periodic orbit
  • Quadratic center
  • Quadratic systems
  • Quadratic vector fields

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