Detecting alien limit cycles near a Hamiltonian 2-saddle cycle

Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie

Research output: Contribution to journalArticleResearchpeer-review

5 Citations (Scopus)


This paper aims at providing an example of a cubic Hamiltonian 2-saddle cycle that after bifurcation can give rise to an alien limit cycle; this is a limit cycle that is not controlled by a zero of the related Abelian integral. To guarantee the existence of an alien limit cycle one can verify generic conditions on the Abelian integral and on the transition map associated to the connections of the 2-saddle cycle. In this paper, a general method is developed to compute the first and second derivative of the transition map along a connection between two saddles. Next, a concrete generic Hamiltonian 2-saddle cycle is analyzed using these formula's to verify the generic relation between the second order derivative of both transition maps, and a calculation of the Abelian integral.
Original languageEnglish
Pages (from-to)1081-1108
JournalDiscrete and Continuous Dynamical Systems
Publication statusPublished - 1 Dec 2009


  • Abelian integral
  • Alien limit cycle
  • Hamiltoman perturbation
  • Limit cycle
  • Planar vector field
  • Transition map
  • Two-saddle cycle


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