# Exercises for Chaos Under Control

## Chapter 4: Linear Dynamics

2. (a) For the bank account depicted by the picture, from a starting balance of \$100 how many years must you wait until your balance first exceeds \$400? (Be fairly careful with your drawing.) What is the effective interest rate? Answer

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

3. You invest \$100 at 10% per year compounded annually. How many years must you wait before your balance exceeds \$200?

(a) Solve the exercise analytically by computing the balance at the end of each year and noting how many years are needed for the balance to exceed \$200. Answer

(b) Solve the exercise graphically by drawing the compounding line corresponding to this interest rate and then using graphical iteration.

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

4.[N] You have held an account at the favorable rate of 10% per year for a number of years. Your current balance is \$1000. How many years before did your balance first become as large as \$500? Solve the exercise graphically. Can you see how to solve this analytically?

5.[N] The current US debt is about \$3 trillion (\$3 x 1012), say. Assuming the debt increases at the steady rate of 10% per year, in how many years will it reach the unimaginable figure of \$1 quadrillion (\$1 x 1015)?

6. Suppose the dynamics of a bank is represented by the compounding line L. Starting from balance A, how many compoundings are necessary for the balance to exceed B? Put appropriate labels on the axes and the two lines shown.

7. Suppose the dynamics of a bank is represented by the compounding line L. Starting from balance A, how many compoundings are necessary for the balance to be less than B? Put appropriate labels on the axes and the two lines shown.

8. Your bank pays an interest rate of 50% compounded annually. On the graph draw (a) coordinate axes, (b) the line Bn+1 = Bn, and (c) the compounding line. Show, by graphical iteration, what the minimum time is for an initial account of \$5 to grow to \$20 under the compounding rules given. (Each small box is \$1 on a side.)

9. In the last decade, the effective interest rate of a bank account in Brazil was -50% compounded annually. Make a graph with coordinate axes Bn (horizontal) and Bn+1 (vertical) and draw on it the line Bn+1 = Bn, and the compounding line. Show, by graphical iteration, the minimum number of years for an initial account of \$300 to shrink to an effective value of \$30 under the compounding rules given. Answer

10. For each of the fixed points a, b, c, and d of the Figure, state whether the point is stable or unstable. Answer