Math 4452 (Peh Ng)

Here are a few examples of how to numerically generate the values of a[n] for a class of discrete dynamical system, using mathematica

Example:  a[n]=.75a[n-1]+16

Input initial condition(s):

a[0]=16

Create a table of a[n] values and name the table P:

P=Table[{n, a[n]=0.75*a[n-1]+16},{n,1,20}]

List the above ordered pairs in a Table Format:

%//TableForm

ListPlot[P,PlotStyle -> {RGBColor[0,0,1],PointSize[0.02]}, AxesLabel -> {"n", " a_n"} ]

Example:  a[n]=.75a[n-1]

a[0]=16

P=Table[{n, a[n]=0.75*a[n-1]},{n,1,20}]

ListPlot[P,PlotStyle -> {RGBColor[0,0,1],PointSize[0.02]}, AxesLabel -> {"n", " a_n"} ]

Example:  The following DDS describes the scenario: Suppose you borrow a[0] amount of money from your friend  Bubba at 1 percent interest per month, and you also  pay Bubba $20 every month. Let a[n] represent the amount of money you still owe  Bubba just after you make your nth payment. Then a DDS is:
 a[n]=1.01a[n-1]-20

a[0]=1000

P=Table[{n, a[n]=1.01*a[n-1] -20},{n,1,20}]

A=ListPlot[P,PlotStyle -> {RGBColor[0,0,1],PointSize[0.02]}, AxesLabel -> {"n", " a_n"} ]

a[0]=3000

P=Table[{n, a[n]=1.01*a[n-1] -20},{n,1,20}]

B=ListPlot[P,PlotStyle -> {RGBColor[1,0,0],PointSize[0.02]}, AxesLabel -> {"n", " a_n"} ]

a[0]=2000

P=Table[{n, a[n]=1.01*a[n-1] -20},{n,1,20}]

CE=ListPlot[P,PlotStyle -> {RGBColor[0,1,0],PointSize[0.02]}, AxesLabel -> {"n", " a_n"} ]

Show[A,B, CE]

Example:  The following is an example of a non-linear DDS , try it out for several different values of a[0] and observe any long-term behavior of the DDS:

a[0]=0.1

P=Table[{n, a[n]=(a[n-1])^2 + 4*(a[n-1])  + 2},{n,1,20}]

%//TableForm