Short: Lyapunov modules for ZoneXplorer
Author: Elena Novaretti (elena [AT] elenadomain [DOT] it)
Uploader: elena elenadomain it
Architecture: ppc-morphos; m68k-amigaos
NO - I'm not back, sorry =) At least... until they will suck money for a dead pltfrm.
But even after that... not likely.
Just, I recently had a chat with an old friend, and he told me he's still playing
with my old good proggy ZX, still appreciating and loving it. And I realized I never
uploaded these additional modules to Aminet - indeed, they were just on my old site,
which I removed since long ! So here're them, if you missed them !
However, for those who may be interested... my old Fractal Gallery is still online
at this "secret" (well, not quite ;) address: www.elenadomain.it/EFG
There's also a new section with a new kind of fractals, which was previously
unpublished. And no, sorry, I can't publish those (and other) modules since they're
definitely not user friendly, they're for experts only (read: you wouldn't enjoy them
INSTRUCTIONS: PLEASE READ !
·What's inside ?
These are two experimental Lyapunov modules for ZoneXplorer (both 68K and PPC MorphOS
compiled mods are included).
The formula used is the classic x->rx(1-x), real numbers, with r alternating
between the plane's x and y values depending on a user-defineable binary mask.
You may copy the Lyap directory to your ZoneXplorer/Modules dir (or just leave it in RAM:, try the modules from there and then decide :-D)
·What about Lyapunov fractals ?
The so called "Lyapunov" fractals were strange, non-2D-coherent, filiform chaotic
pictures computed by iterating a nonlinear formula in the Real domain, like the
original quadratic formula x->rx(1-x), where 'x' is the iteration variable and 'r' a
parameter. I wrote "non-2D-coherent" for the process being 1-dimensional and using
real numbers, but it is "forced" to be 2-dimensional because of the way it is
represented on the 2D plane of parameters: the graphic result won't thus have a
2-dimensional coherent geometry as most other fractals have, where a complex number or
an explicit 2-dimensional process is iterated instead.
Any initial value of 0<x<1 will converge to a same attractor, which can be a single
point, a finite set of points or a fractal set, depending on r. For every x,y point on
the plane, an r is passed at any new iteration which is either r=x or r=y, depending on
an user-specified binary mask, and so on cyclically. Many iterations are allowed, in
order for the iterative dymanic to stabilize on his 'destiny', and a value is then
computed (the so called Lyapunov's exponent) whose value is someway indicative of how
much ordered or chaotic the attractor is. Tricky, ways less "elegant" than conventional
dimensional-coherent dynamics I usually like to explore; but that is, not my invention.
The resulting pictures are however sometimes quite evocative, depicting a magic space
inhabitated by surreal presences resembling ghost ships and magic birds in fly; so
I also decided to implement this kind of proceedings into some ZoneXplorer modules.
I published just a couple of them for now: from an artistic viewpoint, you won't
get so different results by changing the formula, believe me... also, x->rx(1-x), which
is the originally suggested formula for Lyapunov fractals, is QUALITATIVELY identical
to most other similar quadratic functions: using x->x²+r won't lead to qualitatively
different scenarios, so as a plot of z->cz(1-z) on the complex plane will give you
pretty the same structures making up a Mandelbrot set - more or less. OTOH, using
higher degree polynomials, fractionals or transcendent functions usually leads to
total chaos - nothing being worth to play with, be sure.
Please note that these modules, as in my habit, do add extra color to the pictures
which is NOT the result of a pre-computed palette application, but which tries to
exploit the process dynamic to some extent (I didn't follow the traditional way of
computing the Lyapunov exponent straightly), yet trying to keep that "Lyapunov look
These are experimental modules so the way parameters are used may look a bit cryptic.
Map mode: JULIA MODE has to be ALWAYS used and is set by default (meaningless pictures
will be generated otherwise)
Cx, Cy: are treated as integers and get the following meaning:
Cx is Mask and Cy is Mask Length LESS ONE
An OFF bit is for "A" and an ON bit is for "B".
Example: to obtain BBABA you must set Cx = 2+8+16 = 26 and Cy = 5-1 = 4
Example: AAAAAABBBBBB is Cx = 63 and Cy = 11
Thresold: once more controls the color diffusion depth (more like a contrast
Iters: as explained previously, you will need LOTS of them, even thousands, when
wanting more details and for the background "noise" to disappear.
I suggest using as few as needed to navigate, adding more and more to zoom
into littler details, then THE MOST ITERS YOU CAN AFFORD, when you're sure
you want to render your zone in high res (maybe you also want to use the
anti-alias option, but only if your picture is not intended for printing !)
As for all ZoneXplorer modules, the images generated with the included modules cannot
be used for _any commercial or promotional purpose_ without the author's written
permission. In any other cases, you're free to publish or use them the way you want, as
long as you will include a short copyright notice about them being made with a software
which is free but still under my copyright.
© ELENA NOVARETTI 2004-2013 - www.elenadomain.it