# Exercises for Chaos Under Control

## Chapter 6: Chaos Under Control

51.[N] A tent map has parameter value equal to 1.5. At one instant x is
10% greater than the fixed point value, for this tent map. By how much would
you have to change s so that the next value of x would exactly equal that of
the fixed point? Answer

52.[N] A tent map has parameter value equal to 1.5. Suppose s can be changed
by a maximum of ±10% in an attempt to control x near the nonzero fixed point
value. Start with x = 0.7. What sequence of s changes are required to bring
x to within ±5% of the fixed point in the shortest time?
Answer

53. A tent map has parameter value equal to 1.8. (The fixed point for
this map is about 0.643.)

(a) Let the starting value x_{0} be 0.5. Find x_{1},
x_{2}, and x_{3} under this map.
Answer

(b) Now suppose s can be changed to a minimum value of 1.6 or to a maximum
value of 2.0. How close to the fixed point will the output from this map be
after three iterations starting from x_{0} = 0.5 if these controlling
perturbations are appropriately applied at each iteration?
Answer

54.[W] Make a list of phenomena in your everyday life that are possible
examples of chaotic dynamics. For each, speculate on the consequences
(good or bad) of your being able to control chaos with small perturbations.
How about control by others?

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Noise, Music, and Visual Art

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