Maple Worksheet for Iterative Functions

Demonstrating Period of a Cycling Oribt

restart;

Load the following library to assist with plotting:

with(plots):

1. After reading the first part of the web site DeVaney's Fractal Geometry of the Mandelbrot Set Fractal (http://math.bu.edu/DYSYS/FRACGEOM/ ) on iteration, practice some iterations with iterative functions -- see problem 4 in Mandelbrot worksheet:

Enter the starting value for x (the seed).

x := ??;

Define the iteration: x = f(x), where f is the desired iterative function below. Then keep placing cursor anywhere on the line and press enter to do the iterations.

x := ??;

2. We are primarily interested in investigating the fate of the orbit of 0 (the starting seed) under the Mandelbrot iteration NiMvSSJ4RzYiLCYqJEYkIiIjIiIiSSJjR0YlRik= with various complex values for c . As an example we try NiMvSSJjRzYiLCYtSSZGbG9hdEdJKnByb3RlY3RlZEdGKTYkIiQ5IiEiIyEiIiomLUYoNiQiI0NGLCIiIkkiSUdGJUYyRjI=.

c := -1.14+0.24*I;

x := 0; count := 0;

Just keep placing the cursor on the line below and press enter repeatedly: (We'll keep track of the number of iterations in the variable count).

x := x^2+c; count := count + 1;

3. The following defines function that relieves us of the tediousness of the above task, and also allows us to keep a display of each iteration. This is a function called mandeliterate and it uses Maple syntax to generate and print the first 100 iterations, or until the orbit has has moved more than 2.2 from the origin (in which case we know it will escape.)

mandeliterate := proc(c) local z, m, p1; z:=0; for m from 0 to 100 while abs(z) < 2.2 do z := z^2 + c; print(m,z); od; end:

4. We now execute this procedure on the complex number NiMsJiQiJDkiISIjISIiKiYkIiNDRiYiIiIlIklHRitGKw==.

mandeliterate(-1.14+.24*I);

5. Looking at the above output (in particular the last 12 or so iterations, what do we determine to be the fate of the orbit of 0 for the complex number c = NiMsJiQiJVY2ISIkISIiKiYkIiRXI0YmIiIiJSJJR0YrRic=?

6. We can plot the points in the complex plane that that the iteration "settles" down to cycle between in the complex plane by doing a point plot.

p1 := pointplot({[-1.070954542,.1130876363],[-.005845182,.0022234355],[-1.139970778,.2400259928],[.101920897,-.3072452357],[-1.224011766,.1773705800],[.326744481,-.1942073537]}, color=red,axes=BOXED):%;

7. It is helpful to connect the points with lines in the order that they are visited in the iteration -- order is red, green, blue, yellow,magenta, black.

with(plottools):

l1 := line([-1.070954542,.1130876363],[-.005845182,.0022234355],color=red): l2 := line([-.005845182,.0022234355],[-1.139970778,.2400259928],color=green):l3 := line([-1.139970778,.2400259928],[.101920897,-.3072452357],color=blue): l4:= line([.101920897,-.3072452357],[-1.224011766,.1773705800],color=yellow): l5 :=line([-1.224011766,.1773705800],[.326744481,-.1942073537],color=magenta): l6 := line([.326744481,-.1942073537],[-1.070954542,.1130876363],color=black):

display(p1,l1,l2,l3,l4,l5,l6);

8. Exercise: Use the mandeliterate procedure to find the fate of the orbits of 0 for of the following complex c values: a) .25-.5i

b) -.5-.55i

c) -.11 -.7i

d) -.11-.86i

e) -.8 +.4i

Then use the Fractal Microscope (click on coordinate information to show coordinates) to find approximately where these points are in the set (or outside of set) and see how the geometry of antennae or Julia set support what you said about the fate.

mandeliterate(.25-.5*I);