On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

Xin Zhou; Cuiping Li

doi:10.1016/j.jmaa.2009.05.034

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

Xin Zhou^*
, Cuiping Li

^*Corresponding author for this work

School of Mathematical Sciences

Beihang University

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

The finite generators of Abelian integral I (h) = ∮_Γh f (x, y) d x - g (x, y) d y are obtained, where Γ_h is a family of closed ovals defined by H (x, y) = x² + y² + a x⁴ + b x² y² + c y⁴ = h, h ∈ Σ, a c (4 a c - b²) ≠ 0, Σ = (0, h₁) is the open interval on which Γ_h is defined, f (x, y), g (x, y) are real polynomials in x and y with degree 2 n + 1 (n ≥ 2). And an upper bound of the number of zeros of Abelian integral I (h) is given by its algebraic structure for a special case a > 0, b = 0, c = 1.

Original language	English
Pages (from-to)	209-215
Number of pages	7
Journal	Journal of Mathematical Analysis and Applications
Volume	359
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2009.05.034
State	Published - 1 Nov 2009

Keywords

Abelian integral
Hamiltonian system
Picard-Fuchs equation
Weakened Hilbert's 16th problem

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10.1016/j.jmaa.2009.05.034

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title = "On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems",

abstract = "The finite generators of Abelian integral I (h) = ∮Γh f (x, y) d x - g (x, y) d y are obtained, where Γh is a family of closed ovals defined by H (x, y) = x2 + y2 + a x4 + b x2 y2 + c y4 = h, h ∈ Σ, a c (4 a c - b2) ≠ 0, Σ = (0, h1) is the open interval on which Γh is defined, f (x, y), g (x, y) are real polynomials in x and y with degree 2 n + 1 (n ≥ 2). And an upper bound of the number of zeros of Abelian integral I (h) is given by its algebraic structure for a special case a > 0, b = 0, c = 1.",

keywords = "Abelian integral, Hamiltonian system, Picard-Fuchs equation, Weakened Hilbert's 16th problem",

author = "Xin Zhou and Cuiping Li",

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doi = "10.1016/j.jmaa.2009.05.034",

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T1 - On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

AU - Zhou, Xin

AU - Li, Cuiping

PY - 2009/11/1

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N2 - The finite generators of Abelian integral I (h) = ∮Γh f (x, y) d x - g (x, y) d y are obtained, where Γh is a family of closed ovals defined by H (x, y) = x2 + y2 + a x4 + b x2 y2 + c y4 = h, h ∈ Σ, a c (4 a c - b2) ≠ 0, Σ = (0, h1) is the open interval on which Γh is defined, f (x, y), g (x, y) are real polynomials in x and y with degree 2 n + 1 (n ≥ 2). And an upper bound of the number of zeros of Abelian integral I (h) is given by its algebraic structure for a special case a > 0, b = 0, c = 1.

AB - The finite generators of Abelian integral I (h) = ∮Γh f (x, y) d x - g (x, y) d y are obtained, where Γh is a family of closed ovals defined by H (x, y) = x2 + y2 + a x4 + b x2 y2 + c y4 = h, h ∈ Σ, a c (4 a c - b2) ≠ 0, Σ = (0, h1) is the open interval on which Γh is defined, f (x, y), g (x, y) are real polynomials in x and y with degree 2 n + 1 (n ≥ 2). And an upper bound of the number of zeros of Abelian integral I (h) is given by its algebraic structure for a special case a > 0, b = 0, c = 1.

KW - Abelian integral

KW - Hamiltonian system

KW - Picard-Fuchs equation

KW - Weakened Hilbert's 16th problem

UR - https://www.scopus.com/pages/publications/67649866127

U2 - 10.1016/j.jmaa.2009.05.034

DO - 10.1016/j.jmaa.2009.05.034

M3 - 文章

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SN - 0022-247X

VL - 359

SP - 209

EP - 215

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

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