Upper bounds for the number of zeroes for some Abelian integrals

Armengol Gasull; J. Tomás Lázaro; Joan Torregrosa

doi:https://doi.org/10.1016/j.na.2012.04.033

Upper bounds for the number of zeroes for some Abelian integrals

Armengol Gasull, J. Tomás Lázaro, Joan Torregrosa

Department of Mathematics

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

28 Citations (Scopus)

Abstract

Consider the vector field x′=-yG(x,y), y′= xG(x,y), where the set of critical points G(x,y)=0 is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4 we recover or improve some results obtained in several previous works. © 2012 Elsevier Ltd. All rights reserved.

Original language	English
Pages (from-to)	5169-5179
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	75
Issue number	13
DOIs	https://doi.org/10.1016/j.na.2012.04.033
Publication status	Published - 1 Sept 2012

Keywords

Abelian integrals
Chebyshev system
Limit cycles
Number of zeroes of real functions
Weak 16th Hilbert's Problem

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title = "Upper bounds for the number of zeroes for some Abelian integrals",

abstract = "Consider the vector field x′=-yG(x,y), y′= xG(x,y), where the set of critical points G(x,y)=0 is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4 we recover or improve some results obtained in several previous works. {\textcopyright} 2012 Elsevier Ltd. All rights reserved.",

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author = "Armengol Gasull and {Tom{\'a}s L{\'a}zaro}, J. and Joan Torregrosa",

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T1 - Upper bounds for the number of zeroes for some Abelian integrals

AU - Gasull, Armengol

AU - Tomás Lázaro, J.

AU - Torregrosa, Joan

PY - 2012/9/1

Y1 - 2012/9/1

N2 - Consider the vector field x′=-yG(x,y), y′= xG(x,y), where the set of critical points G(x,y)=0 is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4 we recover or improve some results obtained in several previous works. © 2012 Elsevier Ltd. All rights reserved.

AB - Consider the vector field x′=-yG(x,y), y′= xG(x,y), where the set of critical points G(x,y)=0 is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4 we recover or improve some results obtained in several previous works. © 2012 Elsevier Ltd. All rights reserved.

KW - Abelian integrals

KW - Chebyshev system

KW - Limit cycles

KW - Number of zeroes of real functions

KW - Weak 16th Hilbert's Problem

U2 - https://doi.org/10.1016/j.na.2012.04.033

DO - https://doi.org/10.1016/j.na.2012.04.033

M3 - Article

SN - 0362-546X

VL - 75

SP - 5169

EP - 5179

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 13

ER -

Upper bounds for the number of zeroes for some Abelian integrals

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Keywords

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