# 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 x0 be 0.5. Find x1, x2, and x3 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 x0 = 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?