The Feigenbaum Diagram

We use a Feigenbaum diagram to summarize the limiting behavior of iterations. It is a plot of the iterates as a function of the parameter value taken after the iterates have "settled down". The command LogistFeig[] defined below creates the points in a Feigenbaum diagram over the a-interval [min, max] with the parameter a sampled at m equally spaced points and with n iterations plotted after the first k iterations are run.

The Mathematica command Fold takes the last element in the iteration list that FoldList would create. Also, the logistic map, f[x, a] = a x (1 - x) is entered as a compiled function to improve performance.

Input := 

LogistFeig[min_Real, max_Real, k_Integer,
         n_Integer, m_Integer] :=
 Module[{f, a, astep, i, j, x, seed},
    seed = 0.223;
    f = Compile[{x, a}, Evaluate[a x (1 - x)]];
    astep = (max - min)/m;
    Partition[
     Flatten[
      Table[
       a = min + j astep;
       Table[{a,
        Fold[f, seed, Table[a, {k+i}]]}, {i, 1, n}
       ], {j, 0, m}
      ]
     ], 2
    ]
  ]   

Below is a graph of the Feigenbaum diagram for the logistic map with a running from [3, 4].

Input := 

ListPlot[LogistFeig[3., 4., 50, 50, 200]]

Output =

-Graphics-

In the exercises, you will see that the bifurcation diagram appears empty if a > 4. While one might be tempted to think that this means that nothing interesting happens to the dynamics of the map for this range of parameter values, actually, nothing could be further from the truth!