Readme:  Short: Lyapunov modules for ZoneXplorer Author: Elena Novaretti (elena [AT] elenadomain [DOT] it) Uploader: elena elenadomain it Type: gfx/fract Version: 1.0 Requires: gfx/fract/ZoneXplorer.lha Architecture: ppcmorphos; m68kamigaos
PREMISE: 
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quite ;) address: www.elenadomain.it/EFG There's also a new section with a new kind of fractals, which was previously unpublished.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 so much).
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(1x), real numbers, with r alternating between the plane's x and y values depending on a userdefineable binary mask.
·Installation
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, non2Dcoherent, filiform chaotic pictures computed by iterating a nonlinear formula in the Real domain, like the original quadratic formula x>rx(1x), where 'x' is the iteration variable and 'r' a parameter. I wrote "non2Dcoherent" for the process being 1dimensional and using real numbers, but it is "forced" to be 2dimensional because of the way it is represented on the 2D plane of parameters: the graphic result won't thus have a 2dimensional coherent geometry as most other fractals have, where a complex number or an explicit 2dimensional 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 userspecified 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 dimensionalcoherent 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(1x), 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(1z) 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.
·Usage
Please note that these modules, as in my habit, do add extra color to the pictures which is NOT the result of a precomputed 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 and feel"
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 = 51 = 4 Example: AAAAAABBBBBB is Cx = 63 and Cy = 11
Thresold: once more controls the color diffusion depth (more like a contrast control here)
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 antialias option, but only if your picture is not intended for printing !)
COPYRIGHT NOTICE 
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 20042013  www.elenadomain.it 
