Assume we are given the formula:

S(t + 1) = rS(t)

where r 0. We want to investigate the long term behaviors of the system under different conditions as follows:

Case 1: r > 1, S(0) = 0.5

Case 2: r > 1, S(0) = -0.5

Case 3: 0

Case 4: 0

Case 5: -1

Case 6: -1

Case 7: r

Case 8: r

Task1:

At first, use rand() to generate a random number between 1 and 2 as the value for r in case 1. Generate 1000 data points by using S(t + 1) = rS(t), similar to what I showed in class. Plot an X-Y chart for these 1000 data points.

Task2:

At first, use rand() to generate a random number between 1 and 2 as the value for r in case 2. Generate 1000 data points by using S(t + 1) = rS(t), similar to what I showed in class. Plot an X-Y chart for these 1000 data points.

Task3:

At first, use rand() to generate a random number between 0 and 1 as the value for r in case 3. Generate 1000 data points by using S(t + 1) = rS(t), similar to what I showed in class. Plot an X-Y chart for these 1000 data points.

Task4:

At first, use rand() to generate a random number between 0 and 1 as the value for r in case 4. Generate 1000 data points by using S(t + 1) = rS(t), similar to what I showed in class. Plot an X-Y chart for these 1000 data points.

Task5:

At first, use rand() to generate a random number between 0 and -1 as the value for r in case 1. Generate 1000 data points by using S(t + 1) = rS(t), similar to what I showed in class. Plot an X-Y chart for these 1000 data points.

Task6:

At first, use rand() to generate a random number between 0 and -1 as the value for r in case 1. Generate 1000 data points by using S(t + 1) = rS(t), similar to what I showed in class. Plot an X-Y chart for these 1000 data points.

Task7:

At first, use rand() to generate a random number between -1 and -2 as the value for r in case 1. Generate 1000 data points by using S(t + 1) = rS(t), similar to what I showed in class. Plot an X-Y chart for these 1000 data points.

Task8:

At first, use rand() to generate a random number between -1 and -2 as the value for r in case 1. Generate 1000 data points by using S(t + 1) = rS(t), similar to what I showed in class. Plot an X-Y chart for these 1000 data points.

Task 9

Describe and compare behaviors of the system for

(i) r > 1,

(ii) 0

(iii) -1

(iv) r

See picture for hint

A C 1 r= 2 S(O) = 3 B 1.779135 0.5 4 5 Time 6 State 0 0.5 1 =$B$1*B6 2 7 8 9 3 10 4 11 12 13 14 Bodo vou 8 15 16 17 18