Abelian integrals and limit cycles

By Freddy Dumortier, Robert Roussarie. 2006

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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.

Subfields: Biochemistry, Cell Biology, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f9391e69c25c7c0a63402b259afe560d160d70f0