Langford Attractor Chaotic Christmas Decoration

I was looking around for something suitably nerdy that I could use as a Christmas tree decoration, and came across a 1984 paper by William Langford entitled “Numerical Studies of Torus Bifurcations” (https://www.researchgate.net/publication/238282172_Numerical_Studies_of_Torus_Bifurcations) that seemed the very thing.

His work starts with a set of non-linear equations that describe a trajectory on an axisymmetric torus (the first model in the set), and then introduces an increasing small non-axisymmetric perturbation, which leads to the classic bifurcation sequence of periodic orbits with frequency doubling (the next two models, which are the period 4 and 16 solutions), and finally chaos (the final model).  They make a nice sequence of models that illustrate some of the concepts of deterministic chaos, but also look interesting enough to hang on your Christmas tree.

Printing such a three-dimensional trajectory as a long cylindrical structure requires a lot of support in the print.  It is a bit fiddly to remove for the more enclosed models, but is quite manageable with some snips and long-nosed pliers, and a little patience.