Lyapunov fractal

Lyapunov fractals were discovered in the late 1980s[2] by the Germano-Chilean physicist Mario Markus from the Max Planck Institute of Molecular Physiology.

They were introduced to a large public by a science popularization article on recreational mathematics published in Scientific American in 1991.

[5] Therefore, all convergent cycles can be obtained by just shifting the iteration sequence, and keeping the starting value 0.5.

For instance, the Lyapunov fractal for the iteration sequence AB (see top figure on the right) is not perfectly symmetric with respect to a and b.

The sequence string for a n-dimensional fractal has to be built from an alphabet with n characters, e.g. "ABBBCA" for a 3D fractal, which can be visualized either as 3D object or as an animation showing a "slice" in the C direction for each animation frame, like the example given here.

Standard Lyapunov logistic fractal with iteration sequence AB, in the region [2, 4] × [2, 4].
Detail of the Lyapunov fractal in the form of a swallow. Iteration sequence AB, in the region [3.81, 3.87] x [3.81, 3.87].
Generalized Lyapunov logistic fractal with iteration sequence AABAB, in the region [2, 4] × [2, 4].
Generalized Lyapunov logistic fractal with iteration sequence BBBBBBAAAAAA, in the growth parameter region ( A , B ) in [3.4, 4.0] × [2.5, 3.4], known as Zircon Zity .
Animation of a 3D Lyapunov fractal with the sequence ABBBCA