Divergence Diagrams (Extensions of Feigenbaum Diagrams for viewing Cantor Sets.)

Below, we use another method for graphing the limiting behavior of iterations. But, rather than waiting for iterates to "settle down", we determine for every point in the (a, x) plane how many iterations it takes before the iterates leave the x-interval [0, 1] (since all iterates outside of [0, 1] diverge.)

Input := 

GetColor[a_Real, x_Real] :=
 Module[{i, initx, newx},
    i = 0;
    initx = x;
    While[i > -10 && initx < 1,
      i--;
      newx = a*initx*(1 - initx);
      initx = newx;
         ];
    Return[i]
       ]; 

Each point represents a Color or GreyScale for a (a, x) coordinate. We want the smallest numbers (in magnitude) to have the lightest shade, so we count down from 0 since Mathematica makes the largest numbers light.

Input := 

ContourPlot[GetColor[a, x], {a, 4., 5.}, {x, 0., 1.},
 PlotPoints -> {200, 200},
 FrameLabel -> {"a", "x"}]

Output =

-ContourGraphics-