Example 1

Enter the function to iterate, the seed, x0, and the number of iterates, n.

Input := 

g[x_] = f[x, 2.8];
x0 = 0.45;
n = 100;

Create a list of Points on the Orbit Diagram

Input := 

pts = Table[0, {i, 1, 2n + 1}];
pts[[1]] = {x0, g[x0]};
Do[x0 = g[x0];
   pts[[i]] = {x0, x0};
   pts[[i + 1]] = {x0, g[x0]},
   {i, 2, 2 n, 2}];

Plot the Points

Input := 

iterateplot = ListPlot[pts,
 PlotJoined -> True,
 DisplayFunction -> Identity];

Plot the Map along with the Replacement Line

Input := 

mapplot = Plot[{x, g[x]}, {x, 0, 1},
 PlotStyle -> {AbsoluteThickness[1], 
               AbsoluteThickness[2]},
 AspectRatio -> Automatic,
 PlotRange -> {0, 1},
 DisplayFunction -> Identity];

Plot the Orbit Diagram

Input := 

Show[iterateplot, mapplot,
 AspectRatio -> Automatic,
 PlotRange -> {{0, 1}, {0, 1}},
 DisplayFunction -> $DisplayFunction];