# Exercises for Chaos Under Control

## Chapter 4: Limits to Growth

11. (a) Two competing banks offer bonus/service charge accounts. One offers a bonus of \$500 and a service charge rate of 20% (that is, the effective interest rate is -20%), the second a bonus of \$300 and a service charge rate of 10%. Which is the better deal in the long run, if you start with a \$0 balance? How long after starting such accounts, each with a balance of \$0, will it take for the two plans to have equal balances?

(b)[C] Run BANKING/HISTOGRAM to check your answer to part (a). Check also with BANKING/TIMESERIES.

12. Suppose the penalty account line in the Figure has a slope between -1 and +1. What is the fate of any starting balance between \$0 and Bmax in this case? Answer

13. For each of these maps, sketch their graphs, find all fixed points, determine their stability, and describe the time series starting with each of the following points: x0 = -2, x0 = -1, x0 = -1/2, x0 = 0, x0 = 1/2, x0 = 1, and x0 = 2. For each, xn+1 =

14. Repeat exercise 13 for these maps

(a) |xn| + 1

(b) |xn|

(c) |xn| - 1

(d) |xn| - 2

15. Repeat exercise 13 for these maps

(a) 0.5|xn| + 1

(b) 0.5|xn|

(c) 0.5|xn| - 1

(d) 0.5|xn| - 2

16. Repeat exercise 13 for these maps

(a) -|xn| + 2

(b) -|xn| + 1

(c) -|xn|

(d) -|xn| - 1

17. Locate all 2-cycles for the maps of exercise 16. Answer

18. In each of these pictures, circle the fixed points. Label the stable (attracting) fixed points with S; label the unstable (repelling) fixed points with U. Answer

19. In each of these pictures, circle the fixed points. Label the stable (attracting) fixed points with S; label the unstable (repelling) fixed points with U.

20. A weird 1-dimensional dynamical system (the bold lines) has the first return map shown below.

(a) How many fixed points does this system have? In each case say whether they are stable or not. (Why?) Answer

(b) Determine (by doing some graphical experiments) what the attractor for this map is. Explain. Answer

21. (a) Locate the fixed points for each of these six functions and determine the stability of each fixed point.

(b) For the top left and top center functions locate a 2-cycle.