# Exercises for Chaos Under Control

## Chapter 6: Noise, Music, and Visual Art

39.[C] Use MUSIC to play 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 play the next 100 points. Comment on how the "tunes" from the different parameter values sound.

40.[C] Use MUSIC to play 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 play the next 100 points. Comment on how the "tunes" from the different parameter values sound.

41. Play the 4-shade Kelly Game by hand, binning the data into the boxes 0<=xn<1, 1<=xn<2, 2<=xn<3, and 3<=xn<=4. Suppose the signal is periodic with period five, repeating the values 0.5, 1.5, 0.7, 2.5, and 3.5, in that order. Investigate the relations between the driving period and the size of the Kelly picture by plotting the Kelly Picture for

(a) an array with seven columns (vertical lines) and five rows (horizontal lines),

(b) an array with ten columns and five rows, and

(c) an array with fifteen columns and five rows.

Comment on the different patterns arising in these pictures, and on how the patterns might be determined by the relations between the period of the signal and the number of columns of the array.

For Exercises 42-45 run KELLY with the interval 0<=xn<=1 divided into ten evenly spaced bins. (Recall the Program generates a Kelly Picture with fifteen columns and ten rows.)

42.[C] Drive KELLY with 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 and drop the first 100 points. Comment on how the pictures from the different parameter values look. Answer

43.[C] Drive KELLY with 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 and drop the first 100 points. Comment on how the pictures from the different parameter values look. Answer

44.[C] Test sensitive dependence on initial conditions for the logistic map by repeating exercise 42, but taking the initial point to be 0.4. Compare the Kelly Pictures corresponding to the same parameter values.

45.[C] Test sensitive dependence on initial conditions for the tent map by repeating exercise 43, but taking the initial point to be 0.4. Compare the Kelly Pictures corresponding to the same parameter values.

46.[C] Drive KELLY with the real world data of exercise 37. To compare with the IFS picture, use four bins for the Kelly Picture. (Remember, you need 150 data points to generate the Kelly Picture.)

47.[C] Drive KELLY with the real world data of exercise 38. To compare with the IFS picture, use four bins for the Kelly Picture. (Remember, you need 150 data points to generate the Kelly Picture.)

48.[C] Modify Exercise 46 for an eight bin Kelly Picture, using the previous two days' closing prices to generate the eight categories for the bins.

49.[C] Modify Exercise 47 for an eight bin Kelly Picture, using data from a parent company and two offspring companies to generate the eight categories for the bins.

50.(a) What can you conclude from the presence of vertical monochrome bands in a Kelly Picture? Answer

(b) What can you conclude from the presence of horizontal monochrome bands in a Kelly Picture? Answer