Chaos and Fractals

In the last 2 decades scientists have begun to understand that very simple mathematical equations can produce quite complicated behavior. This complicated behavior has acquired the name "chaos". Chaos at first appears to have a random nature, but upon closer examination, it is seen to possess a strange kind of regularity. Patterns repeat as they are examined on an ever finer scale (this repetition is called "fractal"). Chaos occurs in studies of weather, biology, economics, astronomy, medicine, and dynamics (to mention only a few disciplines).

Consider the three coupled linear diferential equations given below, the Lorenz equats:

As you can see, chaos only occurs in at least three dimensions. Those being X, Y, and Z with time sometimes thought of as a fourth dimension. Therefore, in order to appreciate the inricacy of such chaotic attractors, as the lorenz attractor, it is very beneficial to view the trajectories three dimensionally. This is three pictures of a Lorenz attractor as it is being rotated. For this attractor, the paramaters have the values of s = 10, b = 8/3, and r = 25.0. As time increases from 0 to 30, the color changes from blue to orange.

Also, here is a listing of four fractals that you can view. They were obtained from the Macintosh software Mandelzot.

This work was done by Tracy Stauffer and Dr. Scott Hendricks

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