Floris Takens discusses the interpretation of experimental data on the onset of turbulence in terms of strange attractors. He argues that new experimental data, especially from Fenstermacher, Swinney, Gollub, and Benson, should be interpreted using strange attractor theory or may challenge previous conclusions. He presents algorithms to determine whether experimental data, such as turbulence onset, can be attributed to strange attractors, emphasizing that these algorithms should be applied directly to the data, not the power spectrum. He doubts the power spectrum contains the relevant information.
Takens reviews the ideas from his previous work [19], comparing them with those of Landau and Lifschitz [13] regarding flow between rotating cylinders. He notes that the discussion in [19] is not limited to this situation but applies to other cases where ordered dynamics transition to chaos. He also mentions that the current discussion is applicable to these cases.
The Taylor-Couette experiment is described, where fluid motion is studied between two rotating cylinders. For various angular velocities Ω, the velocity at a fixed point is measured over time. In [19], it was proposed that for each Ω, the set of all possible states forms a Hilbert space of divergence-free vector fields. An evolution semi-flow is defined, with an attractor Λ_Ω to which all evolution curves tend as t approaches infinity. The main assumptions in [19] include that the flow on the attractor behaves like an attractor in a finite-dimensional system. Takens justifies these assumptions and uses genericity assumptions, noting that symmetry must hold for systems like Couette flow. In the Landau-Lifschitz picture, the limiting motion is assumed to be quasi-periodic.Floris Takens discusses the interpretation of experimental data on the onset of turbulence in terms of strange attractors. He argues that new experimental data, especially from Fenstermacher, Swinney, Gollub, and Benson, should be interpreted using strange attractor theory or may challenge previous conclusions. He presents algorithms to determine whether experimental data, such as turbulence onset, can be attributed to strange attractors, emphasizing that these algorithms should be applied directly to the data, not the power spectrum. He doubts the power spectrum contains the relevant information.
Takens reviews the ideas from his previous work [19], comparing them with those of Landau and Lifschitz [13] regarding flow between rotating cylinders. He notes that the discussion in [19] is not limited to this situation but applies to other cases where ordered dynamics transition to chaos. He also mentions that the current discussion is applicable to these cases.
The Taylor-Couette experiment is described, where fluid motion is studied between two rotating cylinders. For various angular velocities Ω, the velocity at a fixed point is measured over time. In [19], it was proposed that for each Ω, the set of all possible states forms a Hilbert space of divergence-free vector fields. An evolution semi-flow is defined, with an attractor Λ_Ω to which all evolution curves tend as t approaches infinity. The main assumptions in [19] include that the flow on the attractor behaves like an attractor in a finite-dimensional system. Takens justifies these assumptions and uses genericity assumptions, noting that symmetry must hold for systems like Couette flow. In the Landau-Lifschitz picture, the limiting motion is assumed to be quasi-periodic.