Illuminating the attractor...
In this section we will discover that there is something
dwelling in the logistic function phase-control space which we
created. This thing has the attribute of stability. That means
that it does not disappear or move around as we iterate the
function. It also has a definite size shape and structure. We can
call it a thing because the questions, "How big is
it?", and "How long does it last?", have meaning
for it. This thing may be thought of as a two-dimensional
attractor. We will use the technique of iteration to reveal its
shape and structure.
Recall in the section on Iteration and Attractors we located
an attractor for a specific value of gain. That attractor might
be single valued, multiple valued or chaotic. In phase-control
space we are able to look at the attractor for many values of
gain at one time. This in effect makes the attractor an object
having dimension of phase (y) and gain (g). The shape of this two
dimensional attractor is one of the interesting aspects of the
study of chaos. We will begin thinking of this thing as a
discreet entity but remember that its shape is determined by the
geometry of the iterated function.
To begin let us examine maps for some higher iteration
numbers. First looking at iteration number 1000. You will notice
that the map is quite smooth for low values of gain and it has
some distinct breaks in it. The first of these is at g=1. We have
already noted that for g less than 1, high powers of g are small
and i1000 involves g raised to absurdly high powers (see the
equations in the
section) so that y remains extremely close to 0 for all g<1. Above g=1, up to about g=3,
i1000 increases in a smooth curve. Something happens at g=3 and something else at about g=3.45.
White rectangles will show up
on the screen to direct your attention to some area of the
display. On the next display we have enclosed an area where there
are sudden changes in direction in the plotted line. We will
examine this area again as we add more iterations to the map.
Keep in mind that we are iterating our function a great many
times before we plot any values so if the attractor is single
valued the function should settle down on it by the time any
point is plotted. Further iterations will not change the value of
the function in that case. Run the
Quadratic Iteration i1000
now if you like, otherwise just look at the image.
Next we add iteration number 1001 to the picture. Notice that i1001 lies directly on
top of i1000 all the way up to g=3. Only when the two iterations
result in different y values do you see different lines. At about
g=3.57 some sort of catastrophe evidently occurs. The map becomes
extremely badly behaved above that value of gain. There seems to
be no pattern to the points. Run the
Quadratic Iteration i1000-i1001
In that region of the map which is peppered with points it
looks like disorder reigns. Because of this appearance you will
hear this described as an area of chaos. Remember when we were
looking at the lower numbered iterations one at a time, the map
got increasingly complex near the right hand end as the iteration
number increased. Still for the low numbered iterations the
plotted points fell on some discernable curve. With 1000
iterations the complexity of that curve is enormous but still
there is nothing random about the location of any plotted
In principle you could write the equation for y as a function
of g for the 1000th iteration and calculate each point, even in
the chaotic region. The apparent scatter of points results from
the limitations of the presentation. The complicated tangle of
the map in this area must be presented on a grid of points with
limited resolution. Hence the chaotic appearance. We should
understand chaos to mean simply that there is no easy way based
on the current iteration to predict where the next iteration will
fall. It should not be taken to imply disorder.
There is a large proportion of the map which presents an
orderly appearance. In these areas the iterations apparently
follow some simple rule in spite of the complex equation
describing the behavior of y with respect to g. In the left part
of the map the iterations run along the horizontal axis. Then
they rise in a smooth curve to a certain point. At this point the
iterations split. Take a look in the outlined area. When we
plotted only i1000 only the upper branch of the map was present.
Adding another iteration produced the lower branch.
Think back to the iteration display for the logistic function.
At certain values of gain, those less than 1, the attractor of
the function was 0. From g=1 to g=3 a single valued attractor
dependent on gain was evident. Above g=3 we began to see the
attractor branching in two, then four, then more branches with
the function visiting each branch in turn. Attractors like this
are called periodic attractors. For some values of gain the
function seemed to never settle down. That is what we are
observing here. The logistic function, deeply iterated, settles
at 0 for g<1.
From g=1 to g=3 where
the attractor is single valued, by the time we have iterated 1000
times one more iteration will not change the value of y so i1001
lies precisely on i1000. In general, if we iterate deeply enough
to settle down on the attractor, more iteration does not matter.
In our example, above g=3 where the attractor has two branches
even numbered iterations settle on one of them and odd numbered
iterations settle on the other. What happens then when there are
four branches. Let's add a couple more iterations on our next
display and see. Run the
Quadratic Iterations i1000-i1003
On the next display we plot 256 iterations starting at i1000. You will observe that more
detail is revealed on the phase-control map. In fact we now have enough detail to do
some exploring of the map. Take a look at the
Quadratic Iterations i1000-i1255
We will zoom in on particular areas of the map on the next displays. In later
parts of the program you will have control of the magnification of portions of
the screen. For now we will automatically select a zoom window denoted by the
white rectangles. The first step in magnification will be to look at the window
outlined on this display where one of the attractor branches fragments into chaos.
Within the chaotic region you will see a miniature version of a portion of
the whole map. It is similar to the whole map but not identical.
This is "self-similarity" across scales. Run the
Iterations i1000-i1255 - Zoom 1.
In each of the next several displays we will select a window
and magnify it to full screen on the succeeding display so that
the last map in the series will be a portion of the first map
magnified about 1 million times. If you push the magnification
much beyond this point you may find that small uncertainties in
the numbers being calculated will be amplified to the point that
the map shows artifacts which arise not from the mathematics but
from the computer itself. With some experience you may come to
recognize and ignore these but for now we will not push that far.
Run the Iterations i1000-i1255 - Zoom 2, 3 and 4 displays.
Iterating a function deeply and
displaying it on a phase-control map in effect reveals the shape
of the attractor in that phase-control space. The interesting
features of the attractor for the logistic function occur in the
gain domain from 3 to 4. It is in that area that we will
concentrate our next exploration. The next display in this
section shows this portion of the attractor. Notice that even in
the regions of chaotic behavior, the function remains within well
defined bounds. We will extend our concept of an attractor to
accommodate these regions. Run the
Window of Order in Chaos
As we will see on the next display, the attractor of the logistic function in
phase-control space exhibits areas where both the periodic and chaotic nature
co-exists. The next display shows the region of the attractor where it is
periodic with period 3. This region is bounded on both the left and right by
regions of chaos. In the chaotic region to the left you can see that the density
of points is not uniform, indicating some periodicity in the chaos. Then in the
periodic region to the right of the bifurcations are bands of chaos with period 3.
Window of order Close Up
In the next lesson we will examine the way the attractor
cascades from a single value to multiple single values to
Are there any questions?