The fixed points, 2-cycles, and 4-cycles we have seen are all examples
of attractors of the forward orbit under f. This concept is
the same as we have seen before: the attractor of a sequence of
iterates is the set A of all points of accumulation of the orbit.
The stable set 
of the attractor A is the set of
all 
such that the set of points of accumulation of 
is A. A is a stable attractor if is nonempty
and open. Here we need to quote a fact gained from deeper analysis:
Theorem: For a fixedIf the attractor is a cycle, that is the only attracting cycle., there is at most one stable attractor A, and the stable set of A is dense in I.
The stable attractor of a dynamical system may be discovered, at least
numerically, by simply letting the system run long enough. This is the
key to unveiling the most fantastic fractal yet seen in this course.
For each 
, we choose a seed 
(it doesn't matter which),
and then we compute say 1000 iterates 
. We ignore
what happens early on in this sequence; however, once we reach
, we then record the next 1000 iterates 
,
. By letting the system run for 1000 iterations,
we are guessing that after that much time we are numerically extremely
close to the attractor. Then the next 1000 iterates will run over all
the points in the attractor. If the attractor is a stable 4-cycle, for
instance, the 4 points in that cycle will be recorded 250 times each.
We may now plot the attractors for all in a single diagram:
the cascade diagram (also called the bifurcation
diagram). The horizontal axis measures running from 0 to
4; the vertical axis measures x from 0 to 1. The vertical
cross-section is the attractor of 
. Generally speaking, a
great deal of numerical computation is involved in the creation of
this diagram. Fortunately, FRACTINT already has a beautifully
optimized version of this program under the fractal type
biflambda (the ``bifurcation diagram'' for the ``lambda''
map). The beautiful cascade diagram created by FRACTINT is shown
in this figure. To consult the FRACTINT
notes on the fractal type biflambda, see the following:
FRACTINT notes on biflambda
Figure 69: The cascade diagram
At 
, we can see the bifurcation as a joining of the curves
There is also a ``cascade of bifurcations:'' 2-cycle splitting into 4-cycle, 4-cycle splitting into 8-cycle, etc. This can be better examined by using FRACTINT's zooming mechanisms. For a windowed FRACTINT, simply press and hold the left mouse button to drag out a zoom window. Releasing the button creates a larger view of that part of the fractal.
In particular, let's zoom in on the cascade of bifurcations. In the ``Basic Options'' menu of FRACTINT, you may want to set maxiter to 1000, to obtain a more accurate portrayal of the attractors. The result of one zoom is shown in this figure.
Figure 70: The bifurcation diagram up close
Here is
another zoom to 
, 0.86<x<0.90.
After all the bifurcations from 
-cycles to 
-cycles, we
reach a limiting value of where at last there is an infinite
attractor, around which the iterates bounce endlessly. The end of the
waterfall is commonly considered the beginning of Chaos.
There is clearly a sizeable degree of self-similarity in the waterfall
of bifurcations. In fact, careful measurement of the values of
where the bifurcations occur led Mitchell Feigenbaum to
discover an astonishing constant of nature. The table below records
the values of where bifurcations occur, as measured by
FRACTINT. With FRACTINT, it's possible to record just a few digits of
accuracy. Set the ``filter'' number of iterations quite high to obtain
the best accuracy. Also, in the Windows version of FRACTINT, be sure
to activate the coordinate window to be able to read off the
coordinates of the bifurcations.
 |  |  | |
 | 3.00000 | ||
 | 3.44949 | 0.44949 | |
 | 3.5441 | 0.09461 | 4.75098 |
 | 3.5644 | 0.0203 | 4.66059 |
The ratios of the gaps in the fourth column of Table 8
was discovered by Feigenbaum to have a limit value 4.6692....
Feigenbaum also sought the bifurcation point for other functions
that were ``hump-shaped,'' such as 
and the ``roof'' function
In all cases, the limiting ratio of the gaps between bifurcations turned out to be 4.6692.... The story goes that Feigenbaum was so excited by the universality of his number that he immediately called his mom and related how this number would make him famous. Fascinating historical details may be found in [Gle88]. The proof of one version of the universality of Feigenbaum's number was given in [CE80]. FRACTINT allows the user to vary the function in the bifurcation diagram; try the fractal types bif=sinpi, bifurcation, etc.
Amazingly, the bifurcation diagram shows that there are moments of
tranquility beyond the threshold of chaos. We see sizeable ``gaps''
where the attractor again becomes a finite set. If we zoom in very
carefully, there is a small region where the attractor has exactly 3
points; that is, a stable 3-cycle occurs. The interval of 3-cycles
appears to be 
. Let's try 
to
see. The web diagram is shown in this
figure.
Figure 71: A periodic point of period 3:
and 