Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations

Taoufik Bakri, Hil Gaétan Ellart Meijer, Ferdinand Verhulst

    Research output: Contribution to journalArticleAcademicpeer-review

    1 Citation (Scopus)
    6 Downloads (Pure)

    Abstract

    Persistence and bifurcations of Lyapunov manifolds can be studied by a combination of averaging-normalization and numerical bifurcation methods. This can be extended to infinite-dimensional cases when using suitable averaging theorems. The theory is applied to the case of a parametrically excited wave equation. We find fast dynamics in a finite, resonant part of the spectrum and slow dynamics elsewhere. The resonant part corresponds with an almost-invariant manifold and displays bifurcations into a wide variety of phenomena among which are 2- and 3-tori.
    Original languageUndefined
    Article number10.1007/s00332-009-9045-2
    Pages (from-to)571-596
    Number of pages26
    JournalJournal of nonlinear science
    Volume19
    Issue number5
    DOIs
    Publication statusPublished - Oct 2009

    Keywords

    • MSC-74J30
    • MSC-35B32
    • MSC-35L70
    • MSC-37G15
    • EWI-16423
    • IR-68368
    • Bifurcations
    • Nonlinear waves
    • Resonance
    • Parametric excitation
    • METIS-264106
    • Averaging

    Cite this

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    title = "Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations",
    abstract = "Persistence and bifurcations of Lyapunov manifolds can be studied by a combination of averaging-normalization and numerical bifurcation methods. This can be extended to infinite-dimensional cases when using suitable averaging theorems. The theory is applied to the case of a parametrically excited wave equation. We find fast dynamics in a finite, resonant part of the spectrum and slow dynamics elsewhere. The resonant part corresponds with an almost-invariant manifold and displays bifurcations into a wide variety of phenomena among which are 2- and 3-tori.",
    keywords = "MSC-74J30, MSC-35B32, MSC-35L70, MSC-37G15, EWI-16423, IR-68368, Bifurcations, Nonlinear waves, Resonance, Parametric excitation, METIS-264106, Averaging",
    author = "Taoufik Bakri and Meijer, {Hil Ga{\'e}tan Ellart} and Ferdinand Verhulst",
    note = "10.1007/s00332-009-9045-2",
    year = "2009",
    month = "10",
    doi = "10.1007/s00332-009-9045-2",
    language = "Undefined",
    volume = "19",
    pages = "571--596",
    journal = "Journal of nonlinear science",
    issn = "0938-8974",
    publisher = "Springer",
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    }

    Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations. / Bakri, Taoufik; Meijer, Hil Gaétan Ellart; Verhulst, Ferdinand.

    In: Journal of nonlinear science, Vol. 19, No. 5, 10.1007/s00332-009-9045-2, 10.2009, p. 571-596.

    Research output: Contribution to journalArticleAcademicpeer-review

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    T1 - Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations

    AU - Bakri, Taoufik

    AU - Meijer, Hil Gaétan Ellart

    AU - Verhulst, Ferdinand

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    PY - 2009/10

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    N2 - Persistence and bifurcations of Lyapunov manifolds can be studied by a combination of averaging-normalization and numerical bifurcation methods. This can be extended to infinite-dimensional cases when using suitable averaging theorems. The theory is applied to the case of a parametrically excited wave equation. We find fast dynamics in a finite, resonant part of the spectrum and slow dynamics elsewhere. The resonant part corresponds with an almost-invariant manifold and displays bifurcations into a wide variety of phenomena among which are 2- and 3-tori.

    AB - Persistence and bifurcations of Lyapunov manifolds can be studied by a combination of averaging-normalization and numerical bifurcation methods. This can be extended to infinite-dimensional cases when using suitable averaging theorems. The theory is applied to the case of a parametrically excited wave equation. We find fast dynamics in a finite, resonant part of the spectrum and slow dynamics elsewhere. The resonant part corresponds with an almost-invariant manifold and displays bifurcations into a wide variety of phenomena among which are 2- and 3-tori.

    KW - MSC-74J30

    KW - MSC-35B32

    KW - MSC-35L70

    KW - MSC-37G15

    KW - EWI-16423

    KW - IR-68368

    KW - Bifurcations

    KW - Nonlinear waves

    KW - Resonance

    KW - Parametric excitation

    KW - METIS-264106

    KW - Averaging

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