Chaos

Robert P. Munafo, 1999 Feb 3.

"Chaos theory" or the study of "chaotic" processes is related to the study of fractals because such processes often generate fractal shapes, or will yield a fractal image if a certain attribute is plotted on a graph.

Such chaotic processes usually involve some sort of dynamic process (things changing over time) following seemingly simple rules, usually with no random element but nevertheless showing a seemingly "random" behavior. The iteration algorithm for the Mandelbrot and Julia sets is an example of such a process.

A common technique in the study of chaotic processes is to look at what happens when a "parameter" (such a constant in an iteration formula) is changed. Different values of the parameter will cause different types of behavior, such as convergence to a constant value, divergence to infinity, simple oscillation, more complex oscillation, or "random" non-repeating oscillation. When you make a color-coded plot of what happens as a function of the parameter, you often get a fractal image. This is how the Mandelbrot Set is plotted.

It is almost true when such plots are made that there is a "period-doubling" area — a range of parameters where convergence changes to simple oscillation, then to a type of period-4 oscillation, then to period-8 oscillation, etc. When such areas are found, the ratios of the distances between each doubling and the next approach the Feigenbaum Constant.