Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields

Gavrilov, Lubomir

Annales de l'Institut Fourier, Tome 49 (1999), p. 611-652 / Harvested from Numdam

Résumé

Soit ${\cal A}$ l’espace vectoriel des intégrales abéliennes

I(h)=∫∫{H≤h}R(x,y)dx∧dy,h∈[0,h˜]

où $H(x,y)=(x^2+y^2)/2+\ldots $ est un polynôme réel fixé, $R(x,y)$ est un polynôme réel quelconque, et $\lbrace H\le h\rbrace $ est l’intérieur de l’ovale de $H$ qui contient l’origine et tend vers lui quand $h\rightarrow 0$. Nous démontrons que si $H(x,y)$ est un polynôme quasi-homogène avec des points critiques de Morse, alors ${\cal A}$ est un ${\Bbb R}[h]$-module libre de type fini, dont nous calculons le rang. Nous trouvons les générateurs de ${\cal A}$ dans le cas où $H$ est de degré trois. Ce résultat est ensuite appliqué à l’étude des perturbations polynomiales de degré $n$ des champs de vecteurs hamiltoniens quadratiques réversibles, avec un centre et un point selle. Nous démontrons que, si la fonction de Poincaré-Pontryagin n’est pas identiquement nulle, alors la borne supérieure exacte du nombre de cycles limites dans tout domaine compact du plan est égale à $n-1$.

Abstract

Let ${\cal A}$ be the real vector space of Abelian integrals

I(h)=∫∫{H≤h}R(x,y)dx∧dy,h∈[0,h˜]

where $H(x,y)=(x^2+y^2)/2+\ldots $ is a fixed real polynomial, $R(x,y)$ is an arbitrary real polynomial and $\lbrace H\le h\rbrace $, $h\in [0,\tilde{h}]$, is the interior of the oval of $H$ which surrounds the origin and tends to it as $h\rightarrow 0$. We prove that if $H(x,y)$ is a semiweighted homogeneous polynomial with only Morse critical points, then ${\cal A}$ is a free finitely generated module over the ring of real polynomials ${\Bbb R}[h]$, and compute its rank. We find the generators of ${\cal A}$ in the case when $H$ is an arbitrary cubic polynomial. Finally we apply this in the study of degree $n$ polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle points. We prove that, if the Poincaré-Pontryagin function is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is $n-1$.

Détail

Publié le : 1999-01-01

MR 2000c:34081 | Zbl 0924.58077

DOI : https://doi.org/10.5802/aif.1684

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