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

Gavrilov, Lubomir

Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 611-652.

Soit $𝒜$ l’espace vectoriel des intégrales abéliennes

$$I\left(h\right)=\int {\int }_{\{H\le h\}}R(x,y)dx\wedge dy,h\in [0,\tilde{h}]$$

où $H(x,y)=(x^2+y^2)/2+...$ est un polynôme réel fixé, $R(x,y)$ est un polynôme réel quelconque, et $\{H\le h\}$ est l’intérieur de l’ovale de $H$ qui contient l’origine et tend vers lui quand $h\to 0$. Nous démontrons que si $H(x,y)$ 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 $n-1$.

Let $𝒜$ be the real vector space of Abelian integrals

$$I\left(h\right)=\int {\int }_{\{H\le h\}}R(x,y)dx\wedge dy,h\in [0,\tilde{h}]$$

where $H(x,y)=(x^2+y^2)/2+...$ is a fixed real polynomial, $R(x,y)$ is an arbitrary real polynomial and $\{H\le h\}$, $h\in [0,\tilde{h}]$, is the interior of the oval of $H$ which surrounds the origin and tends to it as $h\to 0$. We prove that if $H(x,y)$ 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 $n-1$.

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