# Exercises for Chaos Under Control

## Chapter 6: Iterated Function Systems

Exercises 26-31 involve playing by hand the four-point square IFS game with corners 1, 2, 3, and 4 arranged counterclockwise, placing 1 in the northeast corner of the square as in Figures 6.26-6.29.

26. Draw the picture if the driving signal is

(a) 2-cycle 1, 2, 1, 2, ... , Answer

(b) 2-cycle 1, 3, 1, 3, ... , Answer

(c) 3-cycle 2, 3, 4, 2, 3, 4, ... . Answer

27. Suppose the data string is uniformly random, but only on the symbols 1, 2, and 3 -- i., e., 4 never occurs. Sketch the picture you expect from running IFS with this string. Answer

28. Suppose the data string is uniformly random, with the restriction that 2 never follows 1. Sketch the picture you expect from running IFS with this string. Answer

29. Suppose the data string is uniformly random, with the restriction that 1 never follows 2. How would the picture be different from that of exercise 28? Answer

30. Suppose the game has been played long enough that every pixel which will be turned on has been turned on. Consider the resulting picture.

(a) What is the largest gap if the only restriction is that 1 never occurs? Answer

(b) What is the largest gap if the only restriction is that the pair 1, 2 never occurs? Answer

(c) What is the largest gap if the only restriction is that the triple 1, 2, 3 never occurs? Answer

(d) What general trend do you think happens here? That is, as the length of the only forbidden sequences increases, what happens to the size of the gaps in the IFS picture? Answer

31. If only 1 and 3 occur in the data string, show the resulting IFS picture lies on the line in the square connecting the vertices 1 and 3 of the square. (Hint: note, the starting point, the middle of the square, also lies on this line segment). Answer

32.[C] Run IFS with 500 points from the following signals:

(a) Gaussian white noise,

(b) Brownian noise, and

(c) pseudo-1/f noise

In this exercise, label the corners of the square 1, 2, 4, 3 clockwise with 1 in the northeast corner of the square. Compare this with Figure 6.26, noting the effect of the ordering of the corners. Can you read the same information out of each picture?

33.[C] Use IFS to analyze 100 points of the tent map in the following parameter ranges: 0.8, 1.1, 1.3, 1.5, 1.7, 1.9. In each case, take the initial point to be 0.5, drop the first 100 points, and plot the next 100 points. Answer

34.[C] Use IFS to analyze 100 points of the logistic map in the following parameter ranges: 0.5, 1.5, 2.5, 3.1, 3.45, 3.55, 3.65, 3.75, 3.85, 3.95. In each case, take the initial point to be 0.5, drop the first 100 points, and plot the next 100 points. Answer

35.[C] For the logistic map, find s values (VERY) near, but less than, the opening of the 3-cycle window. Use IFS for these s-values, and for s = 3.8284 (just inside the opening of the 3-cycle window). Do you see any hint of the 3-cycle window before it opens up?

36.[C] Investigate the effect on IFS plots of sensitive dependence on initial conditions. Use IFS to analyze 100 points of the logistic map for the parameter 3.95 and with initial point 0.5, and dropping the first 100 points. Repeat with initial point 0.4. Describe the similarities and differences of the two plots. Answer

37.[C] Drive IFS with coarsened real-world data. For example, pick a stock whose closing price is available for a long period (at least a few hundred days) and try binning the data like this: 1 if both yesterday's and today's prices closed up (or unchanged), 2 if today's price closed up (or unchanged) and yesterday's closed down, 3 if today's price closed down and yesterday's closed up (or unchanged), and 4 if both today's and yesterday's prices closed down. Try a similar analysis with the performance of your favorite baseball team, if you can get the record for several years. Here "win" corresponds to "closing up," and "lose" corresponds to "closing down."

38.[C] Drive IFS with coarsened real-world data. Consider a pair of stocks (for example, a parent company and one of its children companies) -- say Amalgamated Alligators and Baby Alligator Business -- and bin the data like this: 1 if both A and B close up (or unchanged), 2 if A closes up (or unchanged) and B closes down, 3 if A closes down and B closes up (or unchanged), and 4 if both close down. What would have happened if you had picked two unrelated companies?