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Determining the self-rotation number following a Naimark—Sacker bifurcation in the periodically forced Taylor—Couette flow

  • J. M. Lopez
  • F. Marques

Abstract.

Systems which admit waves via Hopf bifurcations and even systems that do not undergo a Hopf bifurcation but which support weakly damped waves may, when parametrically excited, respond quasiperiodically. The bifurcations are from a limit cycle (the time-periodic basic flow) to a torus, i.e. Naimark—Sacker bifurcations. Floquet analysis detects such bifurcations, but does not unambiguously determine the second frequency following such a bifurcation. Here we present a technique to unambiguously determine the frequencies of such quasiperiodic flows using only results from Floquet theory and the uniqueness of the self-rotation number (the generalization of the rotation number for continuous systems). The robustness of the technique is illustrated in a parametrically excited Taylor—Couette flow, even in cases where the bifurcating solutions are subject to catastrophic jumps in their spatial/temporal structure.

Key words. Floquet theory, self-rotation number, parametric excitation, quasiperiodic flow, Taylor--Couette flow. 

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Copyright information

© Birkhäuser Verlag, Basel, 2000

Authors and Affiliations

  • J. M. Lopez
    • 1
  • F. Marques
    • 2
  1. 1.Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USAUS
  2. 2.Departament de Física Aplicada, Universitat Politècnica de Catalunya, Jordi Girona Salgado s/n, Mòdul B4 Campus Nord, 08034, Barcelona, SpainES

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