\fs28 Lyapunov is a program which plots a fractal space known as a "Lyapunov Space" (hence the odd name). This space is generated by a simple iterated equation, commonly used for predicting populations:\
\fs38 x
\fs34\dn8 n+1
\fs38\dn0 =rx
\fs34\dn8 n
\fs38\dn0 (1-x
\fs34\dn8 n
\fs38\dn0 )\
\fs28 \
where
\fs24
\fs28 "r" is a constant of proportionality. The new twist here, used to generate a Lyapunov space, is that the value of "r" changes during repeated iterations and that the Lyapunov exponent of "x" is of more interest than the actual values "x" takes on. The Lyapunov exponent is approximated by taking an average, using this formula for each "x" generated:\
\fs36 Lyap.[x] = Avg[log
\fs32\dn10 2
\fs36\dn0 (abs(r-2rx
\fs32\dn8 n
\fs36\dn0 ))]\
\fs28 \
The value of "r" alternates between two values, determined by the (x,y) coordinates in a plane. The repetition pattern is predefined and determines what the final picture will look like. To plot the Lyapunov space, a solid "background" color is assigned to all positive Lyapunov exponents, and a shaded color is assigned to the negative exponents. (The "background" color appears in areas where the exponent is unstable, i.e. chaotic.)\
This application implements numerous controls to allow you to change the images generated and study in detail areas of the images. To begin, you select the area of the plane to plot. The viewing area is set up in a normal x-y coordinate system; you give the lower left coordinates of the area to view and the width and height of the area to view. After having drawn an image, if you want to zoom in on an area, just click the mouse down in the center an drag outward until the outline drawn encloses the area you wish to view next. (This works like the Mandelbrot application.)\
Next, you need to select an image depth. This is the number of iterations that will be performed. You will find that 10 iterations takes about 1.5 - 2 minutes to generate an image and 400 iterations takes over an hour. To generate an accurate image, however, 4,000 iterations (or more) are required. This will take in the neighborhood of 9 hours to accomplish. (But it sure looks hot in full color!) I have found that using 50-100 iterations is good to help me locate an area to view. Once I know the area I want to study, I then to the 4,000 iteration image. Finally, you can change the initial value used for "x
\fs20\dn10 0
\fs28\dn0 " in the first iteration. This CAN affect the image, oddly enough! When you're ready to plot the image, just click on the "Go" button.\
In order to color the system, the basic rules applied are this: a positive exponent is chaotic and is painted the background color. A negative exponent is stable; the more negative, the more stable. The "from" color is the color used when closest to zero and the "to" color is used near negative infinity. The contrast slider determines how quickly the shades change from one color to another. Usually values between 1.5 and 5 seem to work best. The "color" button is used to change the colors on a previously colored image - this is MUCH faster than regenerating the entire image!\
Since large Depth value take forever to plot, there is a plot mode switch in the bottom left hand corner. For low Depth values, use line mode - each line is plotted in it's entirety, one at a time, as it is calculated. On values of 4,000 or more, use dot mode for better feedback. This makes the individual pixels appear one at a time as they are generated. (The plotting overhead involved in doing this is trivial when compared to how long it takes to iterate for each dot! It can take seconds to calculate one dot.)\
This gives a basic overview of the program. Some .tiff files are included with this distribution to show sample outputs - in 12 bit color - and instructions for generating them are included in the README file. Of course, you'll want to be inventive and find images that no one has ever seen before... For more information about Lyapunov spaces (and the inspiration for this program) see Scientific American, September 1991, pp. 178-180 for an informative article and other sources of information.\
\fs20 Source code for this program is included in the hopes that (1) some genius might find a way to speed it up since I'm no master C genius type of person and (2) someone can pick up useful ideas for their own future programs. But... here's the legal stuff so that no one gets ripped off:\
\fs18 This software is freeware. You may freely distribute and use it for any purpose as long as the author's name, copyright notice, and this license remain intact. The author is not responsible for any damages, consequential or inconsequential, resulting from the use and distribution of this software. Source code, although provided, may not be used in any future software which is sold for gain without an express written agreement with the author. This software, or any program derived from it, may not be sold for gain. (Only a reasonable distribution fee is permissable.) No warranty, express or implied, is provided with this software.\
\fs20 \
To contact the author, write to: Permanent (home):\
\fs22 Don Yacktman Don Yacktman (Jr.)\
Broadbent Hall #25 2826 N. Elm Lane\
Brigham Young University Arlington Hts., IL, 60004\
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