Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields
Annales de l'Institut Fourier, Volume 49 (1999) no. 2, p. 611-652
Let $𝒜$ be the real vector space of Abelian integrals$I\left(h\right)=\int {\int }_{\left\{H\le h\right\}}R\left(x,y\right)dx\wedge dy,\phantom{\rule{0.277778em}{0ex}}h\in \left[0,\stackrel{˜}{h}\right]$where $H\left(x,y\right)=\left({x}^{2}+{y}^{2}\right)/2+...$ is a fixed real polynomial, $R\left(x,y\right)$ is an arbitrary real polynomial and $\left\{H\le h\right\}$, $h\in \left[0,\stackrel{˜}{h}\right]$, is the interior of the oval of $H$ which surrounds the origin and tends to it as $h\to 0$. We prove that if $H\left(x,y\right)$ is a semiweighted homogeneous polynomial with only Morse critical points, then $𝒜$ is a free finitely generated module over the ring of real polynomials $ℝ\left[h\right]$, and compute its rank. We find the generators of $𝒜$ 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 $n-1$.
Soit $𝒜$ l’espace vectoriel des intégrales abéliennes$I\left(h\right)=\int {\int }_{\left\{H\le h\right\}}R\left(x,y\right)dx\wedge dy,\phantom{\rule{0.277778em}{0ex}}h\in \left[0,\stackrel{˜}{h}\right]$$H\left(x,y\right)=\left({x}^{2}+{y}^{2}\right)/2+...$ est un polynôme réel fixé, $R\left(x,y\right)$ est un polynôme réel quelconque, et $\left\{H\le h\right\}$ est l’intérieur de l’ovale de $H$ qui contient l’origine et tend vers lui quand $h\to 0$. Nous démontrons que si $H\left(x,y\right)$ est un polynôme quasi-homogène avec des points critiques de Morse, alors $𝒜$ est un $ℝ\left[h\right]$-module libre de type fini, dont nous calculons le rang. Nous trouvons les générateurs de $𝒜$ 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 $n-1$.
@article{AIF_1999__49_2_611_0,
author = {Gavrilov, Lubomir},
title = {Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {49},
number = {2},
year = {1999},
pages = {611-652},
doi = {10.5802/aif.1684},
zbl = {0924.58077},
mrnumber = {2000c:34081},
language = {en},
url = {http://www.numdam.org/item/AIF_1999__49_2_611_0}
}

Gavrilov, Lubomir. Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields. Annales de l'Institut Fourier, Volume 49 (1999) no. 2, pp. 611-652. doi : 10.5802/aif.1684. http://www.numdam.org/item/AIF_1999__49_2_611_0/

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