L.S.

Today (25-10-93) I have uploaded the archive LYAP_20B.ARJ in the directory
/pub/msdos/incoming . It contains a program I made myself after reading an
article in the Scientific American about the Lyapunov Space. Mr. Lyapunov was
(or is, I don't recall) a Russian scientist in mathematics who has sort of in-
vented a formula for determining how chaotic the recursive formula x<-rx(1-x)
behaves. By taking the X or Y coordinate of a picture for 'r' in a predefined
pattern (e.g. AB means alternate between X and Y) and converting the result to
a color, you get very nice pictures that can be virtually infinite magnified
like ordinary fractals. In principle it is not a fractal, but it behaves just
like it (and takes much more time to calculate).

This program calculates and displays a Lyapunov Space (and saves the result as
a PCX picture) on any IBM-compatible machine with MS-DOS 3.0+, (S)VGA card
(VESA compatibility needed for resolutions above 320x200) and a 80286 or above.
A coprocessor or 8087 emulator is also needed. Although I have written the pro-
gram in pure assembler and optimised it for speed, the algorithm takes a lot of time (for maximum precision 80000 floating point instructions per pixel) so 
80386 with real copro is highly recommended. You can always set the number of
iterations very low (e.g. 100 instead of 4000) so the result will not be optimal, but come much quicker (with 100 iterations about 40 times quicker; see
LYAPUNOV.DOC for more details), or use a 'draft' mode of 25 iterations just to
see what the picture will look like.

Programs like FractInt now also include something they call 'Lyapunov', but in
fact that is just the basic formula 'AB' (my program can handle ANY formula and
you can even make your own palettes) and I have found my program about three
times faster when usingthe same amount of iterations. A non-copro version will
become available as soon as I have had the time to program an 8087-emulator. I
am also working on a version that saves pictures as GIF instead of PCX, and one
that can have handle a custom resolution instead of the 'standard' (S)VGA
resolutions.

Have fun exploring the Lyapunov Space!

R.vanMeurs@research.ptt.nl (this address will last until 15 december 1993)