Christophe LETELLIER
22/01/2020

**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

For parameter values

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

**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].

**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 R^{2}, *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.