Extension aux cycles singuliers du theoreme de Khovanski-Varchenko

TL;DR

This paper extends the Khovanski-Varchenko theorem to singular cycles, establishing bounds on the number of limit cycles for perturbed Hamiltonian systems with Morse functions and non-vanishing Abelian integrals.

Contribution

It provides a new bound N(d) for the number of limit cycles in perturbed Hamiltonian systems with singular cycles, generalizing previous results to include singularities.

Findings

Bound N(d) depends only on degree d

Limit cycles are bounded for Morse Hamiltonians with non-vanishing Abelian integrals

Results apply to small perturbations of Hamiltonian systems

Abstract

Let dH be a Hamiltonian one form on the real plane, of degre d. We show that, if H is a Morse function, generic at infinity, then there exists a number N(d) depending only on d, such that every small perturbation of dH has at most N(d) limit cycles on the hole real plane, assuming that it's of degre at most d, and that it has a non vanishing Abelian integral along real cycles of dH.