First, the applet itself. It comes in two flavours: with an orthodox coloration, asigning yellow to stable regions and blue to chaotic regions, and a coloration I prefer using green:
Lyapunov Java Applet (orthodox yellow/blue version)
Lyapunov Java Applet (my green version)
I will not explain the mathematical backgrounds here; if you are interested, this is explained in a mathematical explanation page. Instead, I will here only discuss the different parameters of the applet.
To start a computation, hit the [do it!]-button. This will display an excerpt of the AB-plane, a rectangle of the form [x1, x2[ x [y1, y2[. If you want to zoom in, there are two possibilities: you can manually enter the values of x1, x2, y1 and y2 (with (x1, y1) the lower left, (x2, y2) the upper right corners of a rectangle), or you can draw a zoom box with the mouse. To get rid of a box, click again. If you want to return from a zoom to a full tiling, you can manually enter the values 0.0, 3.14159, 0.0 and 3.14159, or you can hit the [return to pi]-button.
A sequence defines a pattern of a’s and b’s, something like ababb. You can abreviate repetitions, that is, write 3a3b instead of aaabbb. You can also use brackets to indicate repetitions, write 3(ab)2(abb) instead of ababababbabb. Anything except digits, brackets, a and b is ignored.
The seed is the initial value of an iteration. Usually, changing this value has only minor effects: it determinates which branches of a stable region are visible and which are invisible.
The function we use to iterate is the function b * sin2(x + r). x is the value that is iterated, r is the alternating value taken from the AB-plane, and b defines the steepness of the function. Usually, the greater b is, the smaller are the stable regions.
The first few iterations are not taken into consideration, since they depend mainly on the seed x. You can determinate how many unused iterations there shall be. 500 would be an acceptable value to obtain exact results, but in most cases, smaller values will still display a result not too unsatisfying.
You can also set the number of used iterations, the number of iterations that are used to compute an approximation of the Lyapunov exponent. 1000 would be a good value, but can slow down things a bit (since this is a Java Applet and not C++), so perhaps it would be wise to try smaller values first.
The max. exp. result is the value that gets the highest color value of the chaotic palette, while 0 gets the lowest color value (we use two color palettes, one for the positive chaotic and one for the negative stable regions). If you are using the green applet version: If the background of the image is very dark, use a smaller value. If the background consists of a uniform bright green, use a higher value. If you are using the yellow/blue applet version: If the background of the image is a uniform blue, use a smaller value. If the background is very dark, use a greater value.
The min. exp. result is the value (multiplied with -1) that gets the highest color value of the stable palette. If there is a lot of white on the image, use a greater value. If there is no white at all on the image, use a smaller value.
To get interesting results, I suggest you first change the sequence, next try different values for b, finally zoom in. Enjoy!
If you are curious about the source code, here it is:
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31.5.2001
Jan Thor
www.janthor.com
jan@janthor.de