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December  2009, 25(4): 1081-1108. doi: 10.3934/dcds.2009.25.1081

Detecting alien limit cycles near a Hamiltonian 2-saddle cycle

Stijn Luca 1, , Freddy Dumortier 2, , Magdalena Caubergh 3, and  Robert Roussarie 4,

1. 

Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek, Belgium
2. 

Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 3590 Diepenbeek
3. 

Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193 Cerdanyola de Vallès, Barcelona, Spain
4. 

Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S., Université de Bourgogne, B.P. 47 870, 21078 Dijon Cedex, France

Received  February 2009 Revised  July 2009 Published  September 2009

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.
Keywords: Abelian integral, Hamiltonian perturbation, transition map., Planar vector field, limit cycle, alien limit cycle, two-saddle cycle.
Mathematics Subject Classification: 34C23, 34C25, 34E10, 34C25, 34C20, 34C0.
Citation: Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081
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