Lorenz Attractor
The first ever attractor of its kind was discovered by Edward Norton Lorenz in 1963 during research about weather behaviour and its predictability^{1}. For a historical introduction to this attractor and chaos theory in general, I can recommend Chaos by James Gleick^{2}. A well researched article by Quanta Magazine gives more insight to the daily work of Lorenz and his colleagues Ellen Fetter and Margaret Hamilton^{3}.
Renders
Differential system:
\[\dot{x} = \alpha\, (y  x)\] \[\dot{y} = x\, (\beta  z)  y\] \[\dot{z} = x\, y  \gamma\, z\]Constants:
\[\alpha = 10\] \[\beta = 28\] \[\gamma = 8/3\]Some Nostalgia
With just a few lines of code you can plot the attractor on BASICable devices. Here, I have tried it on a TI Voyage 200, a model from 2002 with just 188kB RAM and a 128p display:
The program is in 68k TIBasic^{4}, which is a modified version by Texas Instruments and it looks like this:
lorenz()
Prgm
ClrDraw
.1 → x
.1 → y
.1 → z
10 → a
28 → b
8/3 → c
For i,0,1000
a*(yx) → dx
x*(bz)y → dy
x*yc*z → dz
x+dx/100 → x
y+dy/100 → y
z+dz/100 → z
PtOn x,z
EndFor
Pause
DispHome
EndPrgm
If you try this on the same model, the window settings were:
xmin = 82
xmax = 82
xscl = 1
ymin = 10
ymax = 60
yscl = 1

E.N. Lorenz, 1963. "Deterministic nonperiodic flow". J. Atmos. Sci. 20(2). doi:10.1175/15200469(1963)020<0130:DNF>2.0.CO;2. ↩

"Chaos: Making a New Science" by James Gleick (1988) on Goodreads. ↩

"The Hidden Heroines of Chaos" published in 2019 by Quanta Magazine. ↩

A great resource on the language is this forum/wiki. ↩