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Detecting alien limit cycles near a Hamiltonian 2-saddle cycle

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Abstract
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
Planar vector field, Hamiltonian perturbation, limit cycle, Abelian integral, two-saddle cycle, alien limit cycle, transition map

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Chicago
Luca, Stijn, Freddy Dumortier, Magdalena Caubergh, and Robert Roussarie. 2009. “Detecting Alien Limit Cycles Near a Hamiltonian 2-saddle Cycle.” Discrete and Continuous Dynamical Systems 25 (4): 1081–1108.
APA
Luca, S., Dumortier, F., Caubergh, M., & Roussarie, R. (2009). Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 25(4), 1081–1108.
Vancouver
1.
Luca S, Dumortier F, Caubergh M, Roussarie R. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. 2009;25(4):1081–108.
MLA
Luca, Stijn et al. “Detecting Alien Limit Cycles Near a Hamiltonian 2-saddle Cycle.” DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 25.4 (2009): 1081–1108. Print.
@article{8581155,
  abstract     = {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.},
  author       = {Luca, Stijn and Dumortier, Freddy and Caubergh, Magdalena and Roussarie, Robert},
  issn         = {1078-0947},
  journal      = {DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS},
  language     = {eng},
  number       = {4},
  pages        = {1081--1108},
  title        = {Detecting alien limit cycles near a Hamiltonian 2-saddle cycle},
  url          = {http://dx.doi.org/10.3934/dcds.2009.25.1081},
  volume       = {25},
  year         = {2009},
}

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