About

Introduction

Chaotic Dynamical Systems is a mathematical way of attempting to predict something that is viewed as chaotic. Things like the weather, and the stock market are some good examples of chaotic dynamical systems. It was thought at one time, that maybe one function could be used to predict the weather. The problem was that these functions could only predict the near future, but anything in the long distance future would end up being extremely wrong. E. N. Lorenz discovered this concept, which is called sensitive dependence on initial conditions also known as the “Butterfly Effect”. For some more info try looking at: http://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.html

Back in the 1920’s a French mathematician, called Gaston Julia, came up with some complicated quadratic functions. When these functions were magnified over and over again, the image would resemble the original image. This is an example of what fractals are. When you view these images, the color black is usually used to signify that the seed-value was not complex. The other colors, you might see, are usually used to show how fast a seed-value escaped to infinity. So if after 1000 iterations of a function, and if the value is still very low then it would probably get colored black. Now if in the program, the value hit a max, then you would want to view the number of iterations it took to reach this max value. Each of these iterations could be given a different color, which would give you more detail of the patterns in the image.

The Mandelbrot set was later discovered in 1980 from these views of all the possible quadratic Julia sets. The Julia sets are usually of the form z² + c, where different values of c give different shapes. The Mandelbrot set is a way of looking at the function with all possible c values. The Mandelbrot set is consider one of the most beautiful shapes in mathematics, and has been used a lot.
Here are some of the fractals that I have saved from this program:

And here is one pic of the program in action taken from a screen shot:

Download

Requirements/Recommend:

Microsoft Windows 2000/XP
Microsoft .NET Framework 1.1

File:

Click Here to Download

Instructions:

Installation:

Simply unzip the file and Run the [FractalViewer.exe] to run the program.

Running:

Select A Fractal

Select [Fractal] from the menu, and choose one that you would want to see.

Change the Colors

Select [Color] from the menu, then select the type that belongs to the fractal you wish to change. Once you have finished changing the colors, then select [Refresh] ->[Show Changes] or just press (Shift + F5).

Print

Select [File] and then [Page Setup] to setup your printer settings, or it will use the default settings.

To print either select [File]->[Print] or just press (Ctrl + P)

Save as Image

To save the image, select [File] -> [Save Image As...] or just press (Ctrl + S). A pop up box will then be shown. Change the image name and select the type of image you would like to save it as. The choices are Bmp, Emf, Exif, Gif, Icon, Jpeg, MemoryBmp, Png, Tiff, and Wmf. The image will be saved in the same location of the [FractalViewer.exe]

Sources

[Devaney] Devaney, R.L, (1992). “A First Course In Chaotic Dynamical Systems: Theory and Experiment”