Final ExamCOMPLETE

Chaos, Fractals, and Dynamical Systems
Due Thursday, December 16th, by 5:00 pm
May be delivered by hand, email, or fax (610-660-3082).  Please ask for confirmation that your fax/email has been received.

Cover Page:  Honesty Statement.  Print the statement and attach it to your exam.

This exam should represent your own work.  You may not use sources other than your notes, homework, labs, the course web page, pages linked to the course web page, Maple, your calculator, or your textbook.  You can also ask or email me for hints.  If I give a hint to one person, I will link the hint to this page so everyone can see it.

Part I:  Treasure Hunt

Find the best approximations you can to the c-values that produce the following Julia sets.  You need not show work.  Question

Julia A

Julia B
Julia C
Julia D
Julia E
Find the locations of the following enlargements of the Mandelbrot set by approximating the coordinates of the centers of each picture.  You need not show work.  Question

Mandelbrot A
Mandelbrot B
Mandelbrot C
Mandelbrot D
Mandelbrot E  (this is hard! just try to get something that has the same ``flavor'' as this picture)

Use what we've learned about the complex quadratic function and the Fractal Microscope or the other applets linked to the course home page to find the following
  1. A non-real number c for which Qc(z) = z2 + c has an attracting fixed point.
  2. The exact coordinates of the attracting fixed point for the c-value you found in  #1.
  3. A non-real number c for which Qc(z) = z2 + c has an attracting 5-cycle.
  4. The approximate coordinates of the attracting 5-cycle for the c-value you found in #3.  You can use the Maple worksheet on iteration to check your work--enter the complex number 3+2i (for example) as 3 + 2*I.
  5. A non-real number c for which the limit of | Qnc(0) | as n goes to infinity is infinity.  Explain why the orbit of 0 must go to infinity for this c-value.
Part II.  Define the complex function Fc by Fc(z) = cz2.
  1. Find Ki  (the filled Julia set for c = i).  Draw a graph of Ki in the complex plane, and sketch a few typical orbits for points inside, outside, and on the boundary of Ki.
  2. Find all fixed points for Fi(z) = iz2 and determine whether they are attracting, repelling, or neutral.
  3. Show that z0 = 2e4Pi*i/3 lies on a prime 2-cycle for F1/2(z) = z2/2, and classify the cycle as attracting, repelling, or neutral.
  4. Show that, for c not equal to 0, Fc(z) = cz2 always has two fixed points, one of which is attracting and one of which is repelling.
  5. Find the Mandelbrot set for the function Fc(z) = cz2.