An inhomogeneous Picard-Fuchs equation

Stephanus A. van Gils

doi:10.1007/978-94-011-0956-7_27

An inhomogeneous Picard-Fuchs equation

Stephanus A. van Gils

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic

Abstract

In the study of singularities of vector fields on the plane the analysis of perturbations of Hamiltonian systems is crucial. The number of isolated limit cycles in the perturbed system is related to the number of zeros of periods (Abelian integrals). If the Hamiltonian function is algebraic, then the there are finitely many independent periods. They satisfy a matrix linear homogeneous differential equation, the Picard-Fuchs equation. See for instance [BK81]. The average of the perturbed vector field over the periodic orbits of the unperturbed problem can be expressed in those periods.

Original language	English
Title of host publication	Dynamics, Bifurcation and Symmetry
Subtitle of host publication	New Trends and New Tools
Editors	Pascal Chossat
Place of Publication	Dordrecht
Publisher	Kluwer
Pages	333-341
Number of pages	10
ISBN (Electronic)	978-94-011-0956-7
ISBN (Print)	978-94-010-4413-4
DOIs	https://doi.org/10.1007/978-94-011-0956-7_27
Publication status	Published - 1994
Event	NATO Advanced Research Workshop on Dynamics, Bifurcation and Symmetry 1993: New Trends and New Tools - Cargèse, France Duration: 3 Sept 1993 → 9 Sept 1993

Publication series

Name	NATO ASI Series C
Publisher	Kluwer Academic
Volume	437
ISSN (Print)	0377-2071

Conference

Conference	NATO Advanced Research Workshop on Dynamics, Bifurcation and Symmetry 1993
Country/Territory	France
City	Cargèse
Period	3/09/93 → 9/09/93
Other	E.B.T.G. Conference

Keywords

Hamiltonian function
Abelian integral
Unperturbed problem
Monotonicity result
Homoclinic connection

Access to Document

10.1007/978-94-011-0956-7_27

An inhomogeneous Picard-Fuchs equation
van Gils, S. A., 1993, Enschede: University of Twente. 11 p. (Memorandum Faculty of Mathematical Sciences; no. 1185)
Research output: Book/Report › Report › Professional

Cite this

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abstract = "In the study of singularities of vector fields on the plane the analysis of perturbations of Hamiltonian systems is crucial. The number of isolated limit cycles in the perturbed system is related to the number of zeros of periods (Abelian integrals). If the Hamiltonian function is algebraic, then the there are finitely many independent periods. They satisfy a matrix linear homogeneous differential equation, the Picard-Fuchs equation. See for instance [BK81]. The average of the perturbed vector field over the periodic orbits of the unperturbed problem can be expressed in those periods.",

keywords = "Hamiltonian function, Abelian integral, Unperturbed problem, Monotonicity result, Homoclinic connection",

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year = "1994",

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language = "English",

isbn = "978-94-010-4413-4",

series = "NATO ASI Series C",

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T1 - An inhomogeneous Picard-Fuchs equation

AU - van Gils, Stephanus A.

PY - 1994

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N2 - In the study of singularities of vector fields on the plane the analysis of perturbations of Hamiltonian systems is crucial. The number of isolated limit cycles in the perturbed system is related to the number of zeros of periods (Abelian integrals). If the Hamiltonian function is algebraic, then the there are finitely many independent periods. They satisfy a matrix linear homogeneous differential equation, the Picard-Fuchs equation. See for instance [BK81]. The average of the perturbed vector field over the periodic orbits of the unperturbed problem can be expressed in those periods.

AB - In the study of singularities of vector fields on the plane the analysis of perturbations of Hamiltonian systems is crucial. The number of isolated limit cycles in the perturbed system is related to the number of zeros of periods (Abelian integrals). If the Hamiltonian function is algebraic, then the there are finitely many independent periods. They satisfy a matrix linear homogeneous differential equation, the Picard-Fuchs equation. See for instance [BK81]. The average of the perturbed vector field over the periodic orbits of the unperturbed problem can be expressed in those periods.

KW - Hamiltonian function

KW - Abelian integral

KW - Unperturbed problem

KW - Monotonicity result

KW - Homoclinic connection

U2 - 10.1007/978-94-011-0956-7_27

DO - 10.1007/978-94-011-0956-7_27

M3 - Chapter

SN - 978-94-010-4413-4

T3 - NATO ASI Series C

SP - 333

EP - 341

BT - Dynamics, Bifurcation and Symmetry

A2 - Chossat, Pascal

PB - Kluwer

CY - Dordrecht

T2 - NATO Advanced Research Workshop on Dynamics, Bifurcation and Symmetry 1993

Y2 - 3 September 1993 through 9 September 1993

ER -

An inhomogeneous Picard-Fuchs equation

Abstract

Publication series

Conference

Keywords

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Research output

An inhomogeneous Picard-Fuchs equation

Cite this