Gluing Bifurcations in a Dynamically Complicated Extended Flow

Abshagen, J.; Pfister, G.; Mullin, T.

doi:10.1103/physrevlett.87.224501

Cited by 36 publications

(25 citation statements)

References 26 publications

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“…12 The low-dimensional dynamics observed in neighboring ranges of Re are organized through the underlying steady solution structure via Silnikov dynamics as shown by Mullin and Price. 13 More recently, Abshagen, Pfister, and Mullin 14 have shown that gluing bifurcations resulting from Z 2 symmetry broken states are important over a much wider range of parameter space than had previously been appreciated. An explanation of the dynamics which can arise at gluing bifurcations is given by Glendinning, Abshagen, and Mullin.…”

Section: Introductionmentioning

confidence: 99%

Symmetry breaking and multiplicity of states in small aspect ratio Taylor–Couette flow

2002

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We report the results of an experimental and numerical study of the bifurcation structure of steady cellular states in small aspect ratio Taylor-Couette flows. In particular we focus on Z 2 symmetry-breaking bifurcations where asymmetric states which break the mid-plane symmetry arise. We find good quantitative agreement between numerical and experimental results for the bifurcation set. The surprising new observation we make is that the symmetry-breaking bifurcation sequences coalesce in a self-consistent way as the aspect ratio is reduced so that only symmetric steady flows exist for very short cylinders. The implications of this for the onset of disordered motion are discussed.

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Section: Introductionmentioning

confidence: 99%

Symmetry breaking and multiplicity of states in small aspect ratio Taylor–Couette flow

2002

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“…They arise from the merging of a period orbit with a saddle-point if a control parameter is varied and originate in the underlying local bifurcation structure of a nonlinear dynamical system [5]. Examples of homoclinic orbits can be found in many nonlinear physical systems, like, e.g., lasers [6], chemical oscillators [7], electronic circuits [8], and fluid flows [9][10][11][12][13]. In order to understand the organization of the bifurcation structure of a spatial extended nonlinear system an in-depth knowledge of the properties of the first bifurcation from the basic state is crucial [1].…”

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confidence: 99%

“…Examples arise from both imperfect local and imperfect global bifurcations in fluid flows with reflection symmetry (see, e.g., [3,9]). …”

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confidence: 99%

Chaos from Hopf bifurcation in a fluid flow experiment

Langenberg

Pfister

Abshagen³

2004

Phys. Rev. E

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Results of an experimental study of a Hopf bifurcation with broken translation symmetry that organizes chaotic homoclinic dynamics from a T2 torus in a fluid flow as a direct consequence of physical boundaries are presented. It is shown that the central features of the theory of Hopf bifurcation in O(2)-symmetric systems where the translation symmetry is broken are robust and are appropriate to describe the appearance of modulated waves, homoclinic bifurcation, Takens-Bogdanov point, and chaotic dynamics in a fluid flow experiment.

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“…As a result the bifurcations from relative equilibria can be analysed in two steps, describing first the bifurcations associated with the orthogonal dynamics, and then adding the corresponding drift along the rotating wave (Krupa 1990). Subsequent instabilities of rotating waves to quasiperiodic flows have been explored experimentally and found to lead to a multitude of very-low-frequency states via a variety of global bifurcations, the details depending on all three governing parameters (Gerdts et al 1994;von Stamm et al 1996;Abshagen, Pfister & Mullin 2001;Abshagen et al 2005).…”

Section: Introductionmentioning

confidence: 99%

Mode competition of rotating waves in reflection-symmetric Taylor–Couette flow

Abshagen¹,

Lopez²,

Marqués³

et al. 2005

J. Fluid Mech.

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We report on the results of a combined experimental and numerical study on mode interactions of rotating waves in Taylor-Couette flow. Our work shows that rotating waves which originate at a Hopf bifurcation from the steady axisymmetric Taylor vortex flow interact with this axisymmetric flow in a codimension-two fold-Hopf bifurcation. This interaction gives rise to an (unstable) low-frequency modulated wave via a subcritical Neimark-Sacker bifurcation from the rotating wave. At higher Reynolds numbers, a complicated mode interation between stable modulated waves originating at a different Neimark-Sacker bifurcation and a pair of symmetrically related rotating waves that originate at a cyclic pitchfork bifurcation is found to organize complex Z 2 -symmetry breaking of rotating waves via global bifurcations. In addition to symmetry breaking of rotating waves via a (local) cyclic pitchfork bifurcation, we found symmetry breaking of modulated waves via a saddle-nodeinfinite-period (SNIP) global bifurcation. Tracing these local and global bifurcation curves in Reynolds number/aspect ratio parameter space toward their apparant merging point, unexpected complexity arises in the bifurcation structure involving non-symmetric two-tori undergoing saddle-loop homoclinic bifurcations. The close agreement between the numerics and the experiment is indicative of the robustness of the observed complex dynamics.

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Gluing Bifurcations in a Dynamically Complicated Extended Flow

Cited by 36 publications

References 26 publications

Symmetry breaking and multiplicity of states in small aspect ratio Taylor–Couette flow

Symmetry breaking and multiplicity of states in small aspect ratio Taylor–Couette flow

Chaos from Hopf bifurcation in a fluid flow experiment

Mode competition of rotating waves in reflection-symmetric Taylor–Couette flow

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