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

Lubomir Gavrilov

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 2, page 611-652
  • ISSN: 0373-0956

Abstract

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Let 𝒜 be the real vector space of Abelian integrals I ( h ) = ∫ ∫ { H ≤ h } R ( x , y ) d x ∧ d y , h ∈ [ 0 , 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 ≤ h } , h ∈ [ 0 , h ˜ ] , is the interior of the oval of H which surrounds the origin and tends to it as h → 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 [ h ] , 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 .

How to cite

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Gavrilov, Lubomir. "Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields." Annales de l'institut Fourier 49.2 (1999): 611-652. <http://eudml.org/doc/75348>.

@article{Gavrilov1999,
abstract = {Let $\{\cal A\}$ be the real vector space of Abelian integrals\begin\{\}I(h)=\int \int \_\{\lbrace H\le h\rbrace \} R(x,y) dx\wedge dy,\;h\in [0,\tilde\{h\}]\end\{\}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 $n-1$.},
author = {Gavrilov, Lubomir},
journal = {Annales de l'institut Fourier},
keywords = {abelian integrals; limit cycles; Morse critical points; generators of ; Hamiltonian vector fields},
language = {eng},
number = {2},
pages = {611-652},
publisher = {Association des Annales de l'Institut Fourier},
title = {Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields},
url = {http://eudml.org/doc/75348},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Gavrilov, Lubomir
TI - Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 2
SP - 611
EP - 652
AB - Let ${\cal A}$ be the real vector space of Abelian integrals\begin{}I(h)=\int \int _{\lbrace H\le h\rbrace } R(x,y) dx\wedge dy,\;h\in [0,\tilde{h}]\end{}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 $n-1$.
LA - eng
KW - abelian integrals; limit cycles; Morse critical points; generators of ; Hamiltonian vector fields
UR - http://eudml.org/doc/75348
ER -

References

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