Discrete Dynamical Systems and Difference Equations -- Answers

1. • p(n + 1) - p(n) = 0.05 p(n)

   • p(n + 1) / p(n) = 1.05

2. • p(n + 1) - p(n) = 250

   • p(n + 1) / p(n) = 1 + 250/p(n)

3. • p(n + 1) - p(n) = 2.8(1 - .001 p(n)) p(n) - p(n)

   • p(n + 1) / p(n) = 2.8(1 - .001 p(n))

4. • p(n + 1) - p(n) = -0.20 p(n) + 3000

   • p(n + 1) / p(n) = 0.80 + 3000 / p(n)

5. • p(n + 1) - p(n) = [R(p(n)) - 1] p(n)

   • p(n + 1) / p(n) = R(p(n))

6. The table below computes the differences

   p(n + 1) - p(n)

   and the ratios

   p(n + 1) / p(n)

   Notice that the differences seem to be varying around a mean of about 1050 while the ratios appear to be declining. Thus, it looks like the underlying model might be

   p(n + 1) - p(n) = 1050

   with some random variation. Random variation is common with real data.

       n          p(n)   p(n + 1) - p(n)  p(n + 1) / p(n)
   
       1         12000
       2         13097       1097.           1.09
       3         14115       1018.           1.08
       4         15199       1084.           1.08
       5         16202       1003.           1.07
   
       6         17242       1040.           1.06
       7         18292       1050.           1.06
       8         19364       1072.           1.06
       9         20431       1067.           1.06
      10         21498       1067.           1.05
   
      11         22500       1001.           1.05
      12         23573       1073.           1.05
      13         24614       1041.           1.04
      14         25633       1019.           1.04
      15         26671       1037.           1.04
   
      16         27743       1073.           1.04
      17         28805       1062.           1.04
      18         29849       1044.           1.04
      19         30878       1029.           1.03
      20         31880       1002.           1.03
   

7. The table below computes the differences

   p(n + 1) - p(n)

   and the ratios

   p(n + 1) / p(n)

   Notice that the differences seem to be steadily increasing while the ratios seem to be varying around a mean of 1.13. Thus, it looks like the underlying model might be

   p(n + 1) = 1.13 p(n)

   with some random variation.

       n          p(n)   p(n + 1) - p(n)  p(n + 1) / p(n)
   
       1         12000
       2         13455       1455.           1.12
       3         15124       1669.           1.12
       4         17134       2010.           1.13
       5         19455       2321.           1.14
   
       6         21839       2384.           1.12
       7         24531       2692.           1.12
       8         27786       3255.           1.13
       9         31246       3460.           1.12
      10         35140       3893.           1.12
   
      11         39682       4543.           1.13
      12         44805       5123.           1.13
      13         50365       5560.           1.12
      14         56934       6568.           1.13
      15         64849       7915.           1.14
   
      16         72636       7788.           1.12
      17         81367       8730.           1.12
      18         91684      10317.           1.13
      19        103579      11895.           1.13
      20        118072      14494.           1.14
   

8. The table below computes the differences

   p(n + 1) - p(n)

   and the ratios

   p(n + 1) / p(n)

   In addition we have added one new column the ratio

   [p(n + 1) - 68] / [p(n) - 68]

   because it is important to look at the difference between an object's temperature and the ambient (or room) temperature. Notice that this last ratio appears to be varying around a mean of 0.971. Thus, the underlying model here moght be something like

   p(n + 1) - 68 = 0.971 [p(n) - 68]

   with some randon variation.

       n          p(n)   p(n + 1) - p(n)  p(n + 1) / p(n)
   
       1         120.00
       2         118.54       -1.46         0.9879         0.9720
       3         117.06       -1.49         0.9874         0.9705
       4         115.68       -1.37         0.9883         0.9720
       5         114.28       -1.40         0.9879         0.9705
   
       6         112.89       -1.38         0.9879         0.9701
       7         111.58       -1.32         0.9883         0.9707
       8         110.31       -1.27         0.9886         0.9708
       9         109.05       -1.25         0.9886         0.9704
      10         107.82       -1.23         0.9887         0.9700
   
      11         106.65       -1.17         0.9892         0.9707
      12         105.54       -1.11         0.9896         0.9712
      13         104.48       -1.06         0.9900         0.9718
      14         103.40       -1.08         0.9897         0.9704
      15         102.38       -1.02         0.9901         0.9712
   
      16         101.42       -0.97         0.9906         0.9719
      17         100.45       -0.97         0.9905         0.9711
      18          99.49       -0.96         0.9904         0.9704
      19          98.57       -0.92         0.9908         0.9709
      20          97.69       -0.88         0.9911         0.9711