Let me just start by saying that I am aware that lyaponov is not a true fractal, but it pruduces very fractal-like images with very fractal-like math. If you havn't tried them yet, it's worth a try. They give very clean (uncluttered) images with a slightly disturbing fell to them. Many years ago I had a program for the Amiga that made some awesome lyaponov images (although I suspect that most of the effect was due to an effective color scheme), but has been unable to find a decent prog for this on the PC. Any suggestions? /Troberg

DUDE? There is like a ton of freeware links on this page??

Yes, I've seen several (although I must admit that it's been some months since I last looked), but none that produced the nice results I want. I also want a prog that can calculate large images (I'm talking about one or more square meters at the printer's resolution), not that I have the disk space and lots of spare CPU time (three fast dual processor machines that mostly idles...). I thought that someone could recommend a prog worth trying, sparing me the effort of sifting through dozens of progs that's not up to the task. /Troberg

Attached Link: http://ultrafractal.com

suggest you try ultrafractal. here's the link

Thanks, looks great. I'll check it when I get home. The only question is how I can do some sneak prints of huge images on the big inkjet plotter at work... /Troberg

Hey uh troberg dude? you got some art to share with us so we can see what it is yer talking about???I am major lost bro..

Sorry, I got those pictures on a disk that crashed on me a couple of years ago. If I make any good ones I'll post them. Try searching the web for lyaponov, there are some excellent images out there. /Troberg

One of the Mac freeware programs that I tried comes with a Lyapunov folder, but I'll be darned if I can recall which one...? (Of course, that doesn't guarantee that it will do them properly!) The apps are all on my old Mac, which will not talk to my new Mac until we buy a $$ piece of networking hardware....... So I'm unable to look it up for you (and fractalless, myself!!) until I can go out and download the apps again. Did you try a search using Lyapunov as a key word??

Sorry, Mac wont help me. I've invested too much money in PC hardware to change and anyway I'm a programmer so a toy like a Mac is not for me (I need a toy like a cluster of three dual processor PC's...). Of course I've tried searching, but I've yet to find one that produced good results. If you want to try it, the best prog I can recommend is an old prog called Lyaponovia for the Amiga. If you don't have an Amiga or (like me) hasn't used it in almost 10 years, there are some good emulators out there (UAE is my favourite). /Troberg

It's correct that Lyapunov images are not fractals in the classic sense of self iterated recursive images, but are true representations of Chaos theory. In starting to piece together an understanding of a Lyapunov function, I went backwards and started with the Lorenz attractor and Lorenz's work in attempting to analyze Deterministic Dynamical Systems. The Lorenz attractor's visual representation is not very interesting, but a fellow by the name of Shaw went a lot further with what we know as strange attractors. I came to find a distinct relationship between attractors and the Lyapunov image. Whereas an image of the chaotic behavior of an attractor is nothing but lines, a Lyapunov image is the representation of the "surface" of that representation. Attractors have the characteristics of a pendulum, rotor, oscillator or whatever in a dynamical system. In looking at a visual representation of an attractor you see where the lines tend to overlap most often and wander off chaotically. So there are areas of stability where the lines overlap. In a Lyapunov image, since you are looking at a surface representation, the lines that overlap are the dark lines that show the stability of the system. This surface image is essentially a cross-section of the lines mapping an attractor's behavior. A 2 axis image of a 4 axis process. What drives this point home is the fact that The Lyapunov function essentially shows the same as a Poincare map. A slice through the tangled heart of an attractor, removing a two-dimensional section. What's left is the surface image of that slice where the stable intersection of points is a line. Lines of stability as previously mentioned. This surface image is not the same as a fractal that can be zomed into with increasing depth. When zooming into a smaler section of a Lyapunov image you are still on the same plane, just zooming into smaller deatail. The image often "breaks up" in detail but not from visual artifacts but from increasing chaos. The most famous images that have sparked interest in Lyapunov images since 1990 has been the work of Mario Markus. http://perso.wanadoo.fr/charles.vassallo/en/lyap_art/lyapdoc.html, and here is a link with software that should replicate his images, http://www.efg2.com/Lab/FractalsAndChaos/Lyapunov.htm If you happen to be an owner of Paul Carlson's Mind-boggling Fractals software, www.mbfractals.com, there is a free program he provides specifically for Lyapunov images that renders them images in the purest form. His email addy is carlson@mbfractals.com. I have atached a version of Mario Markus' "Zircon Zity" rendered in the program. It is a Logistic equation using the string bbbbbbaaaaaa. A fellow by the name of Francois Blais was very successful with Lyapunov images rendered in Fractint, spanky.triumf.ca/www/fractint/fractint.html, and posted some extraordinary work to alt.binaries.picures.fractals several years ago and again recently. I have not seen any of his posts in 2-3 months, but very likely posting a query to him, via that forum, might produce results. I believe he is with the University of Quebec. Some of Stephen Ferguson's programs, http://www.eclectasy.com/Iterations-et-Flarium24/ produce Lyapunov images, too. Lyapunov Java applet: http://www.janthor.de/Lyapunov/ All of the software shown will allow you to change the colors to suit you, and Fractint probably the best. Given the nature of the image, and what is represented, the color range is limited. Some of the best Lyapunov images done on a large scale are often composite images for more interest. You can find some of these with a www.google.com search.

Thanks! Really good information and an impressive knowledge. I'll try the progs you mentioned later, right now I have some difficulties using my computer since I got attacked last night and scratched up my hands pretty bad (can't rest my wrists on the table...). /Troberg

geez, get better soon!! you're welcome for the info, I like Lyapunov images and that's why I tried to find out what made them tick, so to speak. Why I couldn't zoom in like a fractal. The knowledge really isn't that impressive, just pieces and parts thrown together. (-: There's just not a lot written about the images, what they are or how and why they act as they do. At least from what I found and what I wanted to know about them.