Abelian integrals and limit cycles |
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Authors: | Freddy Dumortier Robert Roussarie |
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Institution: | a Universiteit Hasselt, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek, Belgium b Institut de Mathématique de Bourgogne, UMR 5584 du CNRS, Université de Bourgogne, BP 47 870, 21078 Dijon cedex, France |
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Abstract: | The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation. |
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Keywords: | Planar vector field Hamiltonian perturbation Limit cycle Abelian integral Two-saddle cycle Asymptotic scale deformation |
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