Program 1 - Orbits

This program has three purposes:
	•Plot an orbit versus the iteration number, i.e. plot the pairs (n,(Subscript[x, n])).
	•Plot an orbit versus another orbit, i.e. plot the pairs ((Subscript[x, n]),(Subscript[y, n])).
	•Plot the distance between two orbits versus the iteration number,
	 	i.e. plot the points (n,|(Subscript[x, n])–(Subscript[y, n])|).

f[x_] := 3.2  x (1 - x)

x =  .3; n = 0;

q1 = Table[{n, x, x = f[x], n = n + 1}, {40}];

q2 = q1[[Range[1, 40], Range[1, 2]]];

q3 = ListPlot[q2, PlotStyle -> PointSize[0.02]];

q4 = ListPlot[q2, PlotJoined -> True];

Show[q3, q4];

The last graph clearly indicates that the orbit is periodic of period 2.  Let us now 
change the function.

f1[x_] := 3.79  x (1 - x);

x =  .3; y =  .30001; n = 0;

q5 = 
        Table[{n, Abs[x - y], x = f1[x], y = f1[y], 
            n = n + 1}, {400}];

q6 = q5[[Range[1, 100], Range[1, 2]]];

q7 = ListPlot[q6, PlotStyle -> PointSize[0.02]];

From the above plot it appears that the orbit O(.3) is unstable. To see this in another way we are going to plot one orbit versus the other.

q8 = q5[[Range[1, 400], Range[3, 4]]];

ListPlot[q8, PlotStyle -> PointSize[0.02], 
        PlotRange -> {{0, 1}, {0, 1}}];

When the point is close to the line y=x, the two states are close to each other. In this graph we loose the "history" of the evolution of the system.