INTRODUCTION TO JULIA SETS

Computer Lab

You may remember when we first started looking at dynamical systems, we were concerned about the escape set-- seed values that would fly off to infinity as you iterated your system. Until now, we have deliberately avoided the escape set and used only seed values that produce a bounded sequence. This group of seeds is the opposite of the escape set; you might even call it the "trapped set." The official name for it is the Julia set, named after the French mathematician Gaston Julia.

Today we will go ahead and use those crazy values from the escape set. They get big very quickly and make the computer overflow. The question is how quickly do they do that? Are some quicker than others? We will assign the seeds different colors according to how quickly they escape.

The recursive rule is of this form: z ------> z² + c.

Both z and c are complex numbers, that is, they have a real part and an imaginary part. When you are setting the value of parameter c, you will need to identify one number for the real part and another number for the imaginary part. The same is true when you are picking your seed. A complex number is simply a number with two attributes.

Log on and choose "Complex Numbers" from the applications window. From the main menu of the Complex Numbers program, choose 2) Gather Data. The computer will then ask you to set the value of parameter c. For this experiment, start with c = .1,.2. This means that the real part of c = .1 and the imaginary part = .2. You type it into the computer all on the same line, using a comma between the two parts of the number. Next you need to pick a seed-- say 0, 0. Then let it run. The computer is squaring the number and adding c to it; then taking that result and feeding it back in to get squared again, etc..

After your sequence runs for a while, it may start to converge to a fixed point, or move around in a cycle, or bounce around chaotically, within certain bounds. Any of these behaviors make the seed a member of the Julia set, because it's not escaping to infinity. The seed 0 + 0i certainly belongs to the Julia set, because its sequence converges quickly to a fixed point. The seed .5, .8, however, leads to a sequence that grows without bound. It is therefore a member of the escape set.

Next ask me for your class' special value of c. (B and D periods will have two different values.) Prepare to take notes on the data sheet below. Many of the seeds will be attracted into a cycle, which means they don't escape. To complete our project, we will need a lot of data with a variety of seeds. This means we will divide the labor in the class and then pool our data later. Each class will be making an enormous Julia Set to decorate the bulletin boards in our classroom. There will be a contest between the two classes, and if you work well as a team, the whole class will win valuable prizes.

When you get to this point, call me over and I will assign you a group of seeds to look at. You must WRITE DOWN which seeds are in the Julia set (mark with a "J"), and which ones are in the escape set. For the ones in the escape set, state the number of iterations it takes it to escape. Other people in the class will be relying upon your data, so record it clearly. People can get surly when homework coupons are involved!

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