Jim Belk Cornell University

Homework 3

Due Date: Friday, September 25

For this assignment, let \(f(x) = 4x(1-x)\).

    1. Create a function itin that takes a number \(x_0\) as input and outputs a list of the first 10 symbols in the itinerary of \(x_0\) under \(f\). For example, itin[0.1] should output {L,L,R,L,R,R,R,L,L,R}. You may find the If function helpful for generating each L or R.
    2. Demonstrate your function from part (a) by computing the first 10 symbols in the itinerary for 0.3.
    3. Use your function from part (a) together with the Manipulate command to find a point \(x\in[0,1]\) whose itinerary begins with \(\mathsf{L}\,\mathsf{R}\,\mathsf{L}\,\mathsf{L}\,\mathsf{L}\,\mathsf{R}\,\mathsf{L}\,\mathsf{R}\,\mathsf{R}\,\mathsf{L}\).
    1. Find the point in \([0,1]\) whose itinerary under \(f\) is \(\overline{\textsf{R}}\) (i.e. \(\mathsf{R}\) repeating). Express your answer as a fraction.
    2. Find the points in \([0,1]\) whose itineraries under \(f\) are \(\overline{\textsf{LR}}\) and \(\overline{\textsf{RL}}\), respectively. Express your answers precisely using square roots.
    3. Find the endpoints of the \(\mathsf{LRR}\) interval for \(f\) (shown in Figure 1.12 of the textbook). Your answers must be correct to five decimal places.
    4. Find the point in \([0,1]\) whose itinerary under \(f\) is \(\overline{\textsf{LRR}}\). Your answer must be correct to five decimal places.
    5. Draw cobweb plots showing the cycles you found in parts (b) and (d).
    6. Use the Stability Test for Periodic Points to determine whether these cycles are attracting or repelling.
  3. Let \(n(k)\) denote the number of periodic points of \(f\) of period \(k\), and recall that \[ n(k) \;=\; 2^k - \sum_{d} n(d). \] where \(d\) ranges over all proper divisors of \(k\).
    1. Create a recursively defined function numpoints that takes a number \(k\) as input and outputs \(n(k)\). You will probably find the Sum, Most, and Divisors commands helpful for this.
    2. Make a list of values of \(n(k)\) for \(k\) from \(1\) to \(20\). Does your function agree with our results from class?