3-Dimensional Hopf bifurcation via averaging theory of second order

Jaume Llibre, Amar Makhlouf, Sabrina Badi

Research output: Contribution to journalArticleResearchpeer-review

14 Citations (Scopus)


We study the Hopf bifurcation occurring in polynomial quadratic vector fields in ℝ3. By applying the averaging theory of second order to these systems we show that at most 3 limit cycles can bifurcate from a singular point having eigenvalues of the form ± bi and 0. We provide an example of a quadratic polynomial differential system for which exactly 3 limit cycles bifurcate from a such singular point.
Original languageEnglish
Pages (from-to)1287-1295
JournalDiscrete and Continuous Dynamical Systems
Publication statusPublished - 1 Dec 2009


  • Averaging theory
  • Hopf bifurcation
  • Limit cycle


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