Scroll Attractors via ⌈x⌉

My exploration of things chaotic continues...

Jinhu Lü et al present multi-scroll chaotic attractors based on a set of equations of the form [1]:

where, for n-scroll attractors, the so-called saturated function series f(x) has a staircase-like appearance - albeit with finite slopes between each horizontal step.  Implementation of Lü's f(x) on my computer is not feasible...

...but, I happen to have a ceiling function available: ceiling(x) = ⌈x⌉ gives the smallest integer not less than x. This is not quite what reference [1] uses, but I imagined (correctly) worth a go. In fact, a bit of fiddling on the computer showed me that constants a, b and d can be set to unity, leaving the rather compact:

where c is a positive constant. (Subtracting the 1/2 from x keeps the attractor centered.) The following results are all y vertical, x horizontal (0.5 V/cm).

c = 0.5

c = 0.2

c = 0.1

The c = 0.2 case on paper (with the computer running 1000 times more slowly than when doing the oscilloscope plots) looks like this:

Here's the patching:

Not surprisingly, the floor function ⌊x⌋ also works. In this case adding 1/2 to x, i.e. ⌊x + ½⌋ keeps the resulting attractor centered.

Reference

[1] Jinhu Lu, Guanrong Chen, Xinghuo Yu, H. Leung, ' Generating multi-scroll chaotic attractors via switching control', 5th Asian Control Conference, 20-23 July 2004 pp. 1753 - 1761.