# Exercises for Chaos Under Control

## Chapter 4: The Butterfly Effect

28.[N] (a) Using a calculator, compute the first eight values of the time series starting with x0 = 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 - x0, x'1 - x1, ..., x'8 - x8. What can you conclude?

29.[N] (a) Using a calculator, compute the first six values of the time series starting with x0 = 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 - x0, x'1 - x1, ..., x'8 - x8. 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 x0 = 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 x0 = 0.48 and x'0 = 0.52? What about x0 = 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 xn+1 = 2xn for 0 ² xn ² 1/2 and xn+1 = 2xn - 1 for 1/2 < xn ² 1.

(a) Without using a calculator, find the first ten points of the time series starting with x0 = 1/2, x0 = 1/4, x0 = 1/8, and x0 = 3/4. What do you conclude? Answer

(b) Now find the first ten points of the time series starting with x0 = 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 x0 = 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 x0 = 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.