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Title: | Abelian integrals and limit cycles | Authors: | DUMORTIER, Freddy Roussarie, R. |
Issue Date: | 2006 | Publisher: | Elsevier | Source: | JOURNAL OF DIFFERENTIAL EQUATIONS, 227(1). p. 116-165 | 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. | Keywords: | Planar vector field; Hamiltonian perturbation; Limit cycle; Abelian integral; Two-saddle cycle; Asymptotic scale deformation | Document URI: | http://hdl.handle.net/1942/1565 | ISSN: | 0022-0396 | e-ISSN: | 1090-2732 | DOI: | 10.1016/j.jde.2005.08.015 | ISI #: | 000238729700006 | Category: | A1 | Type: | Journal Contribution | Validations: | ecoom 2007 |
Appears in Collections: | Research publications |
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