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Organizers |
Zeros of Abelian Integrals and Limit Cycles
by
Robert Roussarie
Université de Bourgogne, Laboratoire de Topologie, UMR 5584 CNRS, 9, avenue Alain Savary, BP 47870, F-21078 - Dijon Cedex, France
Coauthors: Freddy Dumortier
Let us consider an analytic vector field unfolding X(\epsilon, [`(\lambda)]), where X(0, [`(\lambda)]) is a fixed Hamiltonian planar vector field. The limit cycles which bifurcate from each closed orbit of this Hamiltonian are given by the zeros of the associated Abelian integral I(h, [`(\lambda)]), at least for generic unfoldings. It is also the case for a singular cycle of the Hamiltonian, through a unique saddle point, i.e. for a saddle connection. If we consider a hyperbolic polycycle of the Hamiltonian with two or more saddle points the situation is much more delicate and it is interesting to look in what extent the Abelian integral remains sufficient to obtain all the limit cycles which can appear for \epsilon\not = 0. We will see that it is not in general the case, because one can break more than one connection. Interesting phenomena are also related to the fact that the different compensators which are linked to each saddle converge toward the same logarithm function when \epsilon tends to 0. This talk is based on a joint paper prepared in collaboration with Freddy Dumortier.
Date received: May 4, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahh-16.