Emergence and bifurcations of Lyapunov manifolds in nonlinear wave equations

Taoufik Bakri, Hil Gaétan Ellart Meijer, Ferdinand Verhulst

Research output: Contribution to journalArticleAcademicpeer-review

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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.",
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volume = "19",
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journal = "Journal of nonlinear science",
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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

TY - JOUR

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

Y1 - 2009/10

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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DO - 10.1007/s00332-009-9045-2

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JO - Journal of nonlinear science

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