Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/10008
Title: DETECTING ALIEN LIMIT CYCLES NEAR A HAMILTONIAN 2-SADDLE CYCLE
Authors: LUCA, Stijn 
DUMORTIER, Freddy 
CAUBERGH, Magdalena 
Roussarie, R.
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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