GS 2003/Math - Collins
Lab 4
Fractal Patterns: Mandelbrot and Julia Sets

In the last lab we explored the behavior of some nonlinear
DEs and (hopefully) discovered that it was not so easy
to classify the bounded/unbounded behavior or sometimes
any type of behavior based on the parameter values.

In this lab we will explore some nonlinear DEs that generate
popular fractal images by systematically exploring and marking
the parameter values. Here's an example of an image you'll be
able to make during this lab.

We'll primarily use the complex DE called the quadratic map: z(n+1) = z(n)^2 + c We'll either vary z(1) in some range of complex values for a fixed value of c, or vary c over a range for some fixed value of z(1). 0. Complex Background All the values we'll be using will be complex numbers, like 1+2i. MATLAB handles complex arithmetic with no extra effort. To enter complex numbers just write 1+2i or 1+2*i or 3.023943-i*72.39 If you have problems it may be because you've accidently changed i from its default value, if so, say clear i and all should be fine. I. Julia Sets For each value of c in the quadratic map, there is a Julia set. The Filled Julia Set is the set of all z(1) such that the values z(n) stay bounded. To create the set we take starting values in the box [-2,2]x[-2,2] and for each one we iterate until either we leave the box, or we reach the limit on the number of iterations. If we reach the limit, we say the point is in the set (and draw) it. Program Info Input consists of c (complex) and m (integer). m denotes how many subdivisions we use in the range of parameter values. For example if m=4 then we use c = a+bi where a, b each comes from the values {-2, -1, 0, 1, 2}. So we'll use 25 different values for c in this case. To make the cool color images, keep track of how many iterations it takes before the iterates leave the box. We'll then color the image according to these values. For example if m=4, then we need to store 25 values in a 5x5 array: it. So if we number the possible values for a (and b) from 1 to 5, then we can store the result of our calculation using the 2nd value for a and the 4th value for b in the (2,4) spot in the array, by it(2,4) = k where k is the number of iterations done. Then to view the resulting picture, just use image(it). (See below about more about color) This is not an impossible program to write, but it does require two for loops and a while loop, so if you are overwhelmed, you can download a version I wrote: julia.m - color filled julia set This is a function called by either julia(c,m) or julia(c,m,[xmin xmax ymin ymax]) where the first instance uses the range [-2,2]x[-2,2], and the second instance uses the range [xmin,xmax]x[ymin,ymax] m is the number of points to divide the ranges into. Try m = 50 to start. It might take to long to compute with larger m. Here's some values for c you might try: -1.5 + 0.2i -0.1 + 0.75i -0.4 + 0.8i 0.28 + 0.53i -0.11 + 0.86i -1.32 0.48 + 0.48i 1.5i -0.5 + 0.57i -0.4 + 0.4i II. Mandelbrot Set There is only one Mandelbrot Set. It is the set of all c values, so that starting with z(1) = 0, the values z(n) remain bounded. The calculation is the same as for the Julia Set, except that c is taken from some range and z(1) is fixed. This program is just a slight modification of the Julia Set program. Again, if you don't want to write the program, you can get a copy here: mandel.m - color mandelbrot set It is a function called by either mandel(m) or mandel(m,[xmin xmax ymin ymax]) where m is the number of points to use in each direction and the range is either the default [-2,2]x[-2,2] or as specified. Again, try small values of m first to see how long it takes. III. (Challenge) Other DEs Any function can be used to generate Julia and Mandelbrot sets. Here's some additional functions you can try: 1. z(n+1) = z(n)^2 + c z(n) 2. z(n+1) = c exp(z(n)) 3. z(n+1) = c sin(z(n)) 4. z(n+1) = z(n)^2/(1 + c z(n)^2) IV. Useful Information You can change the colors by using the command colormap colormapname where colormapname is one of: hsv hot gray bone copper pink white flag lines colorcube jet prism cool autumn spring winter summer If you want to print out a graph, you can just choose print from the file menu. (It will come out in b/w). If you want to save a graph as a jpeg to put on your webpage, type axis off print -djpeg your_filename.jpg If you want another type of file, use help print to get the options. V. Assignment 1. Generate several Julia sets. 2. Generate different views of the Mandelbrot set. 3. Have fun!
ccollins@math.utk.edu
Last Modified: June 14, 2003