The Lorenz Attractor is an excellent and common example of attractors. In fractals, an attractor
is an area that pulls points towards itself, much like the gravity around a star. Those points
will remain chaotic and change orbits unpredictably. However, regardless of how chaotic the
points become, they will always remain in an orbit around the attractor.
Edward Lorenz discovered this attractor during his research into weather patterns. He did this
during the 1960's, before chaos' mainstream acceptance, when the accurate prediction of weather
was thought to be easy. While attempting to simulate weather trends in a three dimensional computer
model, Lorenz accidentally created this fascinating attractor.
The Lorenz attractor is created by iterating equations, just like many other fractals. It is somewhat
unique, though, in that it uses three equations, X, Y, and Z, to make a three dimensional image,
not the two dimensional ones that are more common. The three equations used to create this
fractal are :
xn = xn-1 + d * a * (yn-1 - xn-1)
yn = yn-1 + d * (xn-1 * (c - zn-1) - yn-1)
zn = zn-1 + d * (xn-1 * yn-1 - b * zn-1)
You may have noticed four constants in the above equations. A, B, C, and D all have specific
values, as determined by Lorenz, to produce this attractor. The values are as follows.
Of course, these four constants may be altered to produce different results. The image to the
right was created by altering each value a small amount. As you can see, the resultant image
follows the same shape and form as the Lorenz attractor, but there are noticeable differences
in the actual location of each point. The general form remains; the details change. This behavior
is present in most fractals - attractors included.
The Turbo Pascal program used to generate these images is available for download. The ZIP file
includes the source code, and a DOS compiled executable file. You'll need a PC with at least 1
MB of video RAM to run this program.