The Feigenbaum Tree is the “bifurcation” graph of iterating the function:
f_r(y) = 4 r y (1-y)
r is a constant, 0.25 < r < 1
0 ≤ y ≤ 1
The iteration of f defines a sequence:
y1 = f_r(y0) (1st iteration)
y2 = f_r(y1) (2nd iteration)
y3 = f_r(y2) (3rd iteration)
We want to find out the behavior of the sequence. (with different values of r, and different initial value y0.)
We added the iteration of three points into the previous picture. Recall, that the y-points
are iterated in the vertical direction, because they belong to the same r value. The starting
point in the three cases is, more or less, the point in the middle of that vertical line.
The iterated points keep approaching the branches of the tree. So, for the green parameter
the iterated white points converge to an orbit of period 2. In the cyan range the whitepoints
converge to an orbit of period 4. In the magenta range the white points converge to an orbit
of period 3.
The densely populated parts in between are called “chaotic regions”.
And what do the thick curves in the chaotic parts signify? Clearly there were more of our
iterated points near the thick curves than where the dot pattern is thinner. We discretize
the y-space into pixel-sized points and count, how often these pixelpoints are visited: