An inhomogeneous Picard-Fuchs equation

    Research output: Chapter in Book/Report/Conference proceedingChapterAcademic

    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 languageEnglish
    Title of host publicationDynamics, Bifurcation and Symmetry
    Subtitle of host publicationNew Trends and New Tools
    EditorsPascal Chossat
    Place of PublicationDordrecht
    PublisherKluwer
    Pages333-341
    Number of pages10
    ISBN (Electronic)978-94-011-0956-7
    ISBN (Print)978-94-010-4413-4
    DOIs
    Publication statusPublished - 1994
    EventNATO Advanced Research Workshop on Dynamics, Bifurcation and Symmetry 1993: New Trends and New Tools - Cargèse, France
    Duration: 3 Sep 19939 Sep 1993

    Publication series

    NameNATO ASI Series C
    PublisherKluwer Academic
    Volume437
    ISSN (Print)0377-2071

    Conference

    ConferenceNATO Advanced Research Workshop on Dynamics, Bifurcation and Symmetry 1993
    CountryFrance
    CityCargèse
    Period3/09/939/09/93
    OtherE.B.T.G. Conference

    Keywords

    • Hamiltonian function
    • Abelian integral
    • Unperturbed problem
    • Monotonicity result
    • Homoclinic connection

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  • Cite this

    van Gils, S. A. (1994). An inhomogeneous Picard-Fuchs equation. In P. Chossat (Ed.), Dynamics, Bifurcation and Symmetry: New Trends and New Tools (pp. 333-341). (NATO ASI Series C; Vol. 437). Dordrecht: Kluwer. https://doi.org/10.1007/978-94-011-0956-7_27