Homework 2
Math 323, Spring 2011
Due Date: Thursday, February 10
Instructions: This assignment includes both a Mathematica component and some proofs.
As with the previous homework, you should feel free to work together on this assignment in
groups of two or three.
1. Write a Mathematica notebook to draw cobweb plots for the function f (x) = cx(1−x).
I recommend the following approach:
• Use the Plot command to graph the function f (x) as well as the line y = x.
• Use Table to make a list of points that the cobweb should go through, e.g.
{{0.2,0.2},{0.2,0.64},{0.64,0.64},{0.64,0.9216},...}.
You may need to use a command like Flatten to properly format the list.
• Use the Line command to produce the cobweb. You will need to use Graphics
to make Mathematica display it.
• Use Show to combine the plot and cobweb line together into a single picture.
Use your notebook to draw cobweb plots of the orbit of x0 = 0.01 for c = 2.2, c = 2.8,
and c = 3.2. Make sure to plot at least 20 iterations.
2. Let f (x) = cx(1 − x), where c ∈ (1, 4].
(a) Find a formula for the positive fixed point of f in terms of c. Prove that this fixed
point is attracting for c < 3 and repelling for c > 3.
(b) For c = 7/2, find a period-two orbit for f by solving the equation f 2 (x) = x, and
prove that this orbit is repelling. Does this function have an attracting orbit?
Explain.
(c) Suppose that, for a given value of c, the point 1/2 is periodic under f . In this
case, prove that the corresponding periodic cycle is attracting.
(d) Find a value of c for which 1/2 has period 3, and three values of c for which 1/2
has period 5. (You will need to use Mathematica for this. Give at least three
decimal places in your answer.) Make a cobweb plot showing the orbit of 1/2 in
each case.