Question 23.87
57. You saw in Figure 23.12 on page 966 that a logistic model can result in a stable value, produce cycling among several values, or result in chaos. Other dynamical systems can exhibit similar behavior. Here, we examine the system in which we start with a positive whole number and iterate the following function:
For example, we have .
- Calculate what happens as is applied repeatedly, starting with 133. What do you observe?
- Pick a number different from 133 and different from 1, and iterate repeatedly. What do you observe?
- Why did we exclude 1 in part (b)?
Try some other values. Can you offer a general conclusion? Can you offer an argument why your conclusion is correct?
57.
(a) 133, 19, 82, 68, 100, 1, 1, …. The sequence stabilizes at 1.
(b) Answers will vary.
(c) That would trivialize the exercise!
(d) For simplicity, limit consideration to 3-digit numbers. Then the largest value of for any 3-digit number is . For numbers between 1 and 243, the largest value of is . Thus, if we iterate over and over—say, 164 times—starting with any number between 1 and 163, we must eventually repeat a number, since there are only 163 potentially different results. And once a number repeats, we have a cycle. Thus, applying to any 3-digit number eventually produces a cycle. How many different cycles are there? That we leave you to work out. Hints: (1) There aren’t very many cycles. (2) There is symmetry in the problem, in that some pairs of numbers give the same result; for example, .