Goal: The goal of this lab is to investigate the periods of the bulbs hanging off the Mandelbrot set.
Procedure: Attached to the "main cardioid" in the Mandelbrot set, you see lots of little decorations. These are disks that are directly attached to the main cardioid. These are called the "principal" bulbs on the Mandelbrot set. As we shall see, if you choose any c-value in one of these bulbs, the orbit of 0 always has the same fate: it tends to an attracting cycle of some period. Your job is to determine this period for the principal bulbs (and other bulbs in this set).
The easy way to do this is to use the Mandelbrot Set Iterator, though this applet is quite old and does not work on all browsers. To find the period of a bulb, simply click in its interior (try for the middle); the orbit of 0 as well as the period is displayed. Alternatively, you may use the Mandelbrot/Julia Set Computer applet, which shows you both the orbit of 0 and the corresponding filled Julia set, but does not display the period of the cycle.
1. Find all of the principal bulbs (the bulbs that are directly attached to the main cardioid, not their satellites) that have periods 2 through 8, inclusive. On a sketch of the Mandelbrot set, indicate the relative ordering of these bulbs. Hint: there are 6 period 7 bulbs and 4 period 8 bulbs.
2. Given two principal bulbs that are close to each other (with no larger bulbs in between), how do you find the period of the largest bulb that lies between them?
3. Now go to the period 2 bulb. Find the periods of all of the bulbs hanging directly off this bulb whose period is "analogous" to that of question 1. What does "analogous" mean here? There should be a pattern that is similar in some sense to the pattern you saw in question 1. Provide a similar sketch.
4. Repeat question 3 for the period 3 bulb attached to the upper part of the cardioid.
5. In a brief essay, discuss how all of these baby bulbs are ordered around their parent bulb.
6. For a given c-value in one of the bulbs, you may also compute the corresponding filled Julia set using the Mandelbrot/Julia Set applet. Clicking on a c-value in a bulb in the Mandelbrot set allows you to display the filled Julia set for that value. Your job is to find each of the following filled Julia sets. That is, on a sketch of the Mandelbrot set, indicate where Julia sets 1-4 came from.