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March  2007, 6(1): 1-21. doi: 10.3934/cpaa.2007.6.1

Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle

Magdalena Caubergh 1, , Freddy Dumortier 1, and  Robert Roussarie 2,

1. 

Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 3590 Diepenbeek, Belgium, Belgium
2. 

Université de Bourgogne, I.M.B., U.M.R. 5584 du C.N.R.S., Agoralaan–gebouw D, 21078-Dijon Cedex, France

Received  January 2006 Revised  June 2006 Published  December 2006

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.
Mathematics Subject Classification: Primary: 34C07, 34C08, 34C37, 34E10; Secondary: 34C23, 34E05, 37G2.
Citation: Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure & Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1
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