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1984 Cord toroidal chaos

Christophe LETELLIER
22/01/2020
JPG - 5.8 ko
William F. Langford

In order to investigate some bifurcations which can lead to turbulence, William F. Langford constructed a model designed for producing toroidal chaos [1]. His objective was to investigate with a numerical approach the bifurcations that occur after the first appearance of an invariant torus. The particular interest of this model is that it is three-dimensional. It reads as

PNG - 8 ko

For parameter values

PNG - 3.5 ko

this system produces a toroidal chaotic attractor shown in Fig. 1.

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Fig. 1. Toroidal chaos produced by the Langford system.

A poincaré section (Fig. 2) reveals some foldings characteristic of chaos on a torus, as described in the Curry-Yorke scenario [2].

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Fig. 2. Poincaré section of the toroidal chaotic attractor.

A similar system was used to describe the dynamics observed in a Taylor-Couette flow  [3].

[1] W. F. Langford, Numerical studies of torus bifurcations. In Numerical Methods for Bifurcation Problems, T. Küpper, H. D. Mittelmann & H. Weber (eds), International Series of Numerical Mathematics, 70, 285-295, Birkhäuser, Basel, 1984

[2] J. Curry & J. A. Yorke, A transition from Hopf bifurcation to chaos : computer experiments with maps on R2, Lecture Notes in Mathematics, 668, 48-66, 1978.

[3] T. Mullin, Finite-dimensional dynamics in Taylor-Couette fow, IMA Journal of Applied Mathematics, 46 (1-2), 109-119, 1991.

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