Homework 2
Math 323, Fall 2012
Due Date: Thursday, September 20
For the following problems, let f : [0, 1] → [0, 1] be the logistic map f (x) = cx(1 − x), where
c is a constant in the interval [0, 4].
1. Write Mathematica code that draws cobweb plots for the function f . Your cobweb
plots should include the graph of the function f , the line y = x, and the line segments
of the cobweb, showing at least 20 iterations. The notebook Homework2Commands.nb
illustrates some commands that may be helpful.
Demonstrate your code by drawing cobweb plots of the orbit of x0 = 0.01 for c = 2.2,
c = 2.8, c = 3.2, and c = 4.
2. Consider the case where c = 7/2, so f (x) =
7
x(1 − x).
2
(a) Find the fixed points for f , and use Theorem 1.5 to show that they are both
repelling.
(b) Use the Solve command to find the 2-cycle for f . Express the points of the cycle
as fractions.
(c) Use the Stability Test for Periodic Points to show that the 2-cycle you found in
part (b) is repelling.
(d) Does the function f have an attracting cycle? Justify your answer.
3. (a) Suppose that, for a certain value of c, the point 1/2 is periodic under f . Use the
Stability Test for Periodic Points to prove that, in this case, the orbit of 1/2 must
be attracting.
(b) Use the NSolve command to find a value of c for which 1/2 is periodic with
period 3.
(c) Use NSolve to find thee different values of c for which 1/2 is periodic with period 5.
(d) Draw cobweb plots showing the orbit of 1/2 for each of the values of c you found
in parts (b) and (c).