chaos  complexity  order
Eight equations generating exotic behavior.
Algorithms, program code and graphical output
by Roger Luebeck 2000, 2017

The mathematical intrigues on these pages have, I suppose,
about 1/10,000 the sophistication and complex behavior as
found in such things as Mandelboxes and other 3D fractal
endeavors which are popular today. But this is the level
of mathematical endeavor I'm capable of, and I do enjoy
creating my own mathematical entities, doing my own
explorations, and making my own discoveries. If you like
the stuff you see here, and have yet to discover the world
of fractals, just do a youtube search on Mandelboxes.
The first two sections below are my own creations. The rest
of the sections I merely added my own investigations to,
especially the logistic map and Sierpinski.
Sections 1, 2, 3 and 4 are by far the most interesting.
1. Balloting
Iteration of a balloting paradigm, using random number
generator. It grew out of my analysis of the 2000
Gore  Bush Florida balloting. Profound patterns
amidst chaos.
2. Simple trig
I played a hunch and came up with a pair of repeating
trig equations, with offsetting scalar multipliers
for each equation. Unbelievable patterns. (See top of
this page and click "Simple trig" to see many more.)
3. Logistic map (LM) (logistic difference equation)
X = r * X * (1  X)
Mathematician Paul Stein called the complexity of
this iterated equation "frightening".
Iterating this equation produces regions of distinct
values, involving period doubling, as well as regions
of chaos.
On the LM page, you'll find bifurcation diagrams
at various scales showing the depth of this equation.
You'll also find the equation graphed parabolically,
superimposed onto a straight line graph and the
bifurcation diagram. The LRLRR patterning and
L:R ratios are examined.
Finally, you'll find circular plots of the equation,
revealing patterns amidst the chaos.
4. Sine and pi (no link called for)
X = r * Sin (pi * X)
Here's what my investigations turned up:
Profoundly 
Iterating this equation generates a bifurcation diagram
visually indistinguishable from the one generated by the
above logistic map (LM), though the actual values are much
different. With LM, the valid values for r range from
0 to 4 and valid values for initial X range from 0 to 1.
With the sine pi equation, r can range from 0 to
infinity, though there is only chaos beyond r = 57.29578.
Initial X can range from 0 to infinity, though the graph
flips upside down when initial X exceeds 57.29578. It
flips right side up again when initial X is double that
value, with the flipping occuring at each multiple of
57.29578, which equals 360/(2 pi). And just as with
the LM, one can generate segments of a parabola by
plotting X vs previous X.
