The Hopf-van der Pol system: Failure of a homotopy method

Hil Gaétan Ellart Meijer, T. Kalmár-Nagy

    Research output: Contribution to journalArticleAcademicpeer-review

    6 Citations (Scopus)
    25 Downloads (Pure)

    Abstract

    The purpose of this article to provide an explicit example where continuation based on the homotopy method fails. The example is a one-parameter homotopy for periodic orbits between two well-known nonlinear systems, the normal form of the Hopf bifurcation and the van der Pol system. Our analysis shows that various types of obstructions can make approximation over the whole range of the homotopy parameter impossible. The Hopf-van der Pol system demonstrates that homotopy methods may fail even for seemingly innocent systems.
    Original languageUndefined
    Pages (from-to)323-328
    Number of pages6
    JournalDifferential equations and dynamical systems
    Volume20
    Issue number3
    DOIs
    Publication statusPublished - 2012

    Keywords

    • EWI-21066
    • Periodic orbits
    • IR-81397
    • Homotopy
    • METIS-296420
    • Global bifurcations

    Cite this

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    abstract = "The purpose of this article to provide an explicit example where continuation based on the homotopy method fails. The example is a one-parameter homotopy for periodic orbits between two well-known nonlinear systems, the normal form of the Hopf bifurcation and the van der Pol system. Our analysis shows that various types of obstructions can make approximation over the whole range of the homotopy parameter impossible. The Hopf-van der Pol system demonstrates that homotopy methods may fail even for seemingly innocent systems.",
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    The Hopf-van der Pol system: Failure of a homotopy method. / Meijer, Hil Gaétan Ellart; Kalmár-Nagy, T.

    In: Differential equations and dynamical systems, Vol. 20, No. 3, 2012, p. 323-328.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - The Hopf-van der Pol system: Failure of a homotopy method

    AU - Meijer, Hil Gaétan Ellart

    AU - Kalmár-Nagy, T.

    N1 - eemcs-eprint-21066

    PY - 2012

    Y1 - 2012

    N2 - The purpose of this article to provide an explicit example where continuation based on the homotopy method fails. The example is a one-parameter homotopy for periodic orbits between two well-known nonlinear systems, the normal form of the Hopf bifurcation and the van der Pol system. Our analysis shows that various types of obstructions can make approximation over the whole range of the homotopy parameter impossible. The Hopf-van der Pol system demonstrates that homotopy methods may fail even for seemingly innocent systems.

    AB - The purpose of this article to provide an explicit example where continuation based on the homotopy method fails. The example is a one-parameter homotopy for periodic orbits between two well-known nonlinear systems, the normal form of the Hopf bifurcation and the van der Pol system. Our analysis shows that various types of obstructions can make approximation over the whole range of the homotopy parameter impossible. The Hopf-van der Pol system demonstrates that homotopy methods may fail even for seemingly innocent systems.

    KW - EWI-21066

    KW - Periodic orbits

    KW - IR-81397

    KW - Homotopy

    KW - METIS-296420

    KW - Global bifurcations

    U2 - 10.1007/s12591-011-0091-5

    DO - 10.1007/s12591-011-0091-5

    M3 - Article

    VL - 20

    SP - 323

    EP - 328

    JO - Differential equations and dynamical systems

    JF - Differential equations and dynamical systems

    SN - 0971-3514

    IS - 3

    ER -