The Road to Chaos is Filled with Polynomial Curves

by Richard D. Neidinger and R. John Annen III

American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640-653

The following color figures supplement the section on Roots and Mandelbrot Buds. Click each image to view an enlargement. See below for a rough explanation.
The Mandelbrot Set.

A zoom around the period-5 bud.

In each image, the color indicates the period of the attractor in the corresponding component.

First, recall the usual (not above) Mandelbrot set in the complex plane. A complex number c is in the Mandelbrot set if the iteration of z^2+c (beginning with zero) remains bounded. It is in a (hyperbolic) component of period n if the iteration attracts to a peroidic orbit of period n. These components are the disks budding off of the Mandelbrot set or cardioids for mini-Mandelbrot sets. The "center" of each of these components occurs where the iteration, starting with zero, is itself periodic; so that the nth iterate is zero. (Called a superattracting root in the paper.)

In the algorithm that produced these images, pixels are given color n whenever the nth iterate has modulus less than 0.2. This forms colored "disks" around centers where the nth iterate is zero. The colored disks are not the components themselves but simply mark a corresponding component. In most cases, the parameter 0.2 is large enough that the disk covers the corresponding (largest covered) component of the Mandelbrot set. One exception is the main cardioid in which the attractor has period one, corresponding to the colored disk in the "center" of the cardioid component.

Look for numerical patterns in the (periods of) disks of descending size around the main cardioid in the first image and also on the five "spokes" in the largest decoration in the second image.

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Last updated on May 29, 2007.