Homework #12

Reading: pg. 549-602

1. Iterate the map x(k+1)=2*x(k) mod 1 for several choices of x(0) chosen from [0,1]. Use both a computer and a calculator. What is the forward orbit of a typical x(0) chosen from the [0,1]? Explain your results.

2. The solution of the recurrence relation x(k)=x(k-1)+x(k-2) (the Fibonacci relation) for x(0)=1, x(1)=p is given by x(k)=p^k where p=(1-sqrt(5))/2. (This is a special case of Binet's formula.) Compute x(2),...x(100) using both the recurrence relation and the formula. Plot your results. Comment.

3. The logistic map (also called the quadratic iterator; see pg.585 and pg. 593 of the text) with parameter a=4 was used as a pseudo-random number generator for a while. Experiment with it; does it seem like it would be a good one?

4. Find all fixed points of the logistic map as a function of the parameter a. Plot the location of the nonconstant fixed point as a function of a as a varies over (0,5].

5. For a=.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, pick a value near the nonzero fixed point of the logistic map and iterate it several times. Describe the results.

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