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

Magdalena Caubergh; Freddy Dumortier; Robert Roussarie

doi:10.3934/cpaa.2007.6.1

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2007, Volume 6, Issue 1: 1-21. Doi: 10.3934/cpaa.2007.6.1

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

1.
Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 3590 Diepenbeek
2.
Université de Bourgogne, I.M.B., U.M.R. 5584 du C.N.R.S., Agoralaan–gebouw D, 21078-Dijon Cedex

Received: December 31, 2005

Revised: May 31, 2006

Published: February 2007

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Abstract

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.

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Mathematics Subject Classification: Primary: 34C07, 34C08, 34C37, 34E10; Secondary: 34C23, 34E05, 37G20.

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