Here we present three interactive applets all dealing with the dynamics
of iterations of the complex function f(z) = z^2 + c, where c is a complex
parameter. We are assuming that you have a little familiarity with orbits,
the Mandelbrot Set and Julia Sets. If you are not familiar with these
topics, see below.
In the applet Orbits you may choose from 16 values of c which were
chosen to illustrate some of the possible different behaviors such as
attracting fixed points, repelling fixed points, attracting periodic points,
repelling periodic points and neutral fixed points. Or you may enter your
own choice of c. Then you may click on a spot in the complex plane and
see the orbit of that point under iteration of z^2 + c.
The applet The Mandelbrot Set allows you to zoom in on points in the
Mandelbrot Set. You may also use this applet to find interesting
values of c to use in the Orbits Applet. You may use the two applets
together to test your knowledge of the periods of cycles found using the
c-value found in the various bulbs on the Mandelbrot Set. The applet
Julia Setsallows you to select a point in the Mandelbrot Set and see
the corresponding Julia Set. You may also use this applet in conjunction
with the Orbits applet and compare the Julia Set for a given value of c
with the orbits of points under iteration of z^2 + c.

There is an excellent introduction to the subject of the Mandelbrot Set
and Julia Sets on the web site of
The Dynamical Systems and Technology Project at Boston University.
By Robert L. Devaney

1. In the applet Orbits, you may select the number of colors, 1-16 and 19. Experiment with a different number of colors for each of the pre-selected parameters p and q. Using only one color first, see if you can decide whether f(z) = z^2+c for that value of c = p + iq has an attracting cycle of some period. For example, if it appears that there is an attracting cycle of period 9, as in the second choice offered for p and q, try using 9 colors.
2. In the applet Orbits, choose one of the pre-selected values for c = p+iq and look at the orbits. Then try your own values for p and q moving the point just slightly away from the pre-selected value and notice how the orbits change.
3. In the applet Orbits start with p = -0.4, q = 0.532 and the number of colors = 5. Let q stay fixed at 0.532 and let p decrease to -0.51 in steps of -.01, looking at the orbits each time you decrease p.
4. See if you can find a period 25 attracting cycle. Since there are no more than 20 colors, what number of colors should you use to best view a period 25 cycle?
5. Zoom in to an interesting area in the Mandelbrot Set, record the value of the center point c = p+iq, and then enter those values for p and q in the applet Orbits and examine some of the orbits for that p and q. Try points from the various bulbs, from the antennae, and as close to the boundary as you can get.
6. Find an interesting Julia Set from the applet Julia Sets, record the values of p and q, and then enter those values in the applet Orbits and examine the orbits. For example, using this method I found p = -0.734, q = 0.167 to yield extremely interesting orbits using 19 colors.
7. Send me at aburns@liu.edu other interesting exercises that make use of these applets and I will post them on this page.

 References
1. Devaney, Robert L., A First Course in Chaotic Dynamical Systems, Addison-Wesley, 1998
2. Mandelbrot, Benoit B., The Fractal Geometry of Nature, W.H. Freeman and Co., 1983
3. Peitgen, H.-O. and P.H. Richter, The Beauty of Fractals, Springer-Verlag, 1986