Detecting alien limit cycles near a Hamiltonian 2-saddle cycle
Stijn Luca Freddy Dumortier Magdalena Caubergh Robert Roussarie
Discrete & Continuous Dynamical Systems - A 2009, 25(4): 1081-1108 doi: 10.3934/dcds.2009.25.1081
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
Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle
Magdalena Caubergh Freddy Dumortier Robert Roussarie
Communications on Pure & Applied Analysis 2007, 6(1): 1-21 doi: 10.3934/cpaa.2007.6.1
It is known that perturbations from a Hamiltonian 2-saddle cycle $\Gamma $can produce limit cycles that are not covered by the Abelian integral, even when the Abelian integral is generic. These limit cycles are called alien limit cycles. In this paper, extending the results of [6] and [2], we investigate the number of alien limit cycles in generic multi-parameter rigid unfoldings of the Hamiltonian 2-saddle cycle, keeping one connection unbroken at the bifurcation.
keywords: Planar vector field Abelian integral asymptotic scale deformation. two-saddle cycle limit cycle Hamiltonian perturbation
Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Lienard systems
Magdalena Caubergh Freddy Dumortier Stijn Luca
Discrete & Continuous Dynamical Systems - A 2010, 27(3): 963-980 doi: 10.3934/dcds.2010.27.963
The paper deals with the cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. We generalize the results in [1] (case $m=1$), providing a substantially simpler and more transparant proof than the one used in [1].
keywords: Liénard equation; limit cycle; heteroclinic connection; cyclicity.

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