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|Title:||DETECTING ALIEN LIMIT CYCLES NEAR A HAMILTONIAN 2-SADDLE CYCLE||Authors:||LUCA, Stijn
|Issue Date:||2009||Publisher:||AMER INST MATHEMATICAL SCIENCES||Source:||DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 25(4). p. 1081-1108||Abstract:||This paper aims at providing and 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.||Notes:||[Luca, Stijn; Dumortier, Freddy] Univ Hasselt, B-3590 Diepenbeek, Belgium. [Caubergh, Magdalena] Univ Autonoma Barcelona, Dept Matemat, E-08193 Barcelona, Spain. [Roussarie, Robert] Univ Bourgogne, Inst Math Bourgogne, UMR 5584, CNRS, F-21078 Dijon, France.||Keywords:||Planar vector field; Hamiltonian perturbation; limit cycle; Abelian integral; two-saddle cycle; alien limit cycle; transition map||Document URI:||http://hdl.handle.net/1942/10008||ISSN:||1078-0947||e-ISSN:||1553-5231||DOI:||10.3934/dcds.2009.25.1081||ISI #:||000271091200001||Category:||A1||Type:||Journal Contribution||Validations:||ecoom 2010|
|Appears in Collections:||Research publications|
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