28.[N] (a) Using a calculator, compute the first eight values of the
time series starting with x_{0} = 0.01 for the tent map with s = 2.

(b) Repeat (a) for a starting value of x'_{0} = 0.02.

(c) Compute the difference between the corresponding entries in the times
series for (a) and (b). That is, compute x'_{0} - x_{0},
x'_{1} - x_{1}, ..., x'_{8} - x_{8}.
What can you conclude?

29.[N] (a) Using a calculator, compute the first six values of the time series
starting with x_{0} = 0.48 for the tent map with s = 0.5.

(b) Repeat (a) for a starting value of x'_{0} = 0.46.

(c) Compute the difference between the corresponding entries in the times
series for (a) and (b). That is, compute x'_{0} - x_{0},
x'_{1} - x_{1}, ..., x'_{8} - x_{8}.
What can you conclude?

(d) What can you conclude comparing the part (c) of this exercise with part (c) of exercise 28?

30.[N] (a) Using a calculator, compute the first three values of the time series
starting with x_{0} = 0.49 for the tent map with s = 2.

(b) Using a calculator, compute the first three values of the time series starting
with x'_{0} = 0.51 for the tent map with s = 2.

(c) Comparing (a) and (b), what can you conclude?

(d) What would happen if you had started with x_{0} = 0.48 and
x'_{0} = 0.52? What about x_{0} = 0.47 and x'_{0} = 0.53?
Can you detect a pattern?

(e) Do the conclusions of parts (c) and (d) depend on taking s = 2?

31.[N] Define a map x_{n+1} = 2x_{n} for
0 ² x_{n} ² 1/2 and x_{n+1} = 2x_{n} - 1 for
1/2 < x_{n} ² 1.

(a) Without using a calculator, find the first ten points of the time series
starting with x_{0} = 1/2, x_{0} = 1/4, x_{0} = 1/8,
and x_{0} = 3/4. What do you conclude? Answer

(b) Now find the first ten points of the time series starting with
x_{0} = 1/3. As in part (a), do the computation by hand.
Do you want to change the conclusion you made in part (a)?
Answer

(c) Repeat (b) for x_{0} = 1/5.
Answer

(d) Using a calculator, compute the first twenty points of the time series.
Since you're using a calculator, you'll have to use a decimal version of 1/3.
Try x_{0} = 0.333. How do you reconcile this answer with the answer of (b)?

32.[N] Show the map of exercise 31 exhibits sensitive dependence on initial conditions. In your analysis, start with two points near 0, say 0.0010 and 0.0011. What happens to the distance between the corresponding points of the time series for the first five iterates?

33.[W] The popular media contain many references to the Butterfly Effect, often stating or implying that "small causes can produce big consequences." Find such a reference (other than Jurassic Park) and comment on whether the author's usage is really an example, or just a misunderstanding.

Return to Chapter 4 Exercises

Return to Chapter 4 Exercises: The Tent Map

Go to Chapter4 exercises: Strange Attractors

Return to Chaos Under Control