Variation from repetition...
In this section we will begin to get into the heart of the
matter. There are only two new displays involved but we will work
them to death.
This idea of iteration is fundamental
to the exploration of chaos. We will practice iteration in a
variety of ways but for now let's focus on the results of
iterating the logistic function.
This might represent the fraction of some
maximum theoretical population of animals with a limited food
supply. Think back to the shape of this curve. For small values
of x, y increased. For larger values of x, y decreased. We will
demonstrate in the remainder of this program that iteration of
even quite simple functions produces immensely complicated
In the last section we iterated this function several times
over a few values of x. (Review
New y by Feedback
In the present case we will iterate a few times over
all the points in our domain at once. This in effect produces a
graph of the function itself for the zeroth iteration. For the
first iteration we stop at each sample value of x in the domain,
calculate a y, substitute that value for x in the function and
run it again. Only the final value of y is plotted above the
Think of the effect that the maneuvers described above will
have on the graph as compared to the graph of the original
function. The function we are dealing with returns small numbers
for small and large numbers and returns large numbers for
mid-domain values. It may be thought of as mapping a straight
line onto an arch, as though our segment of the real number line
was stretched into a new shape by the function. Now if we take
the numbers on this arch as input to the function, which is what
happens in iteration, we will find the arch mapped onto a new
Let us try to reason out the shape to expect for the first
iteration of the logistic function. It should be zero where the
original function curve was zero and it should be low where the
original function curve was high. It should be high where the
original function curve was medium. The logistic function was low
on the ends and high in the middle, passing through medium twice.
The graph of the first iterate should be lower in the middle,
zero on the ends and higher between the ends and the middle.
Exactly the same hand waving
argument holds for successive iterations so the graph of each
iterate should be flatter on the top and steeper on the sides
than the previous one. This is exactly what you will find for
gains less than about 3. As iteration number increases, the curve
approaches a rectangular shape indicating that whatever the
starting value of x greater than 0 and less than 1, y approaches
a single value. For gains greater than 3 something different
happens and for reasons we will explore later there are two or
more possible limiting values of y. Run the
You should see on this display how iteration develops a
rectangular curve of y vs. x.
Are there any questions?
Now we will
run the display
with some new parameters. You should notice that the shape of any particular
iterate of the function depends strongly on the value of gain but
in general they begin to take on a square wave appearance as the
iteration number goes up. Beyond a certain gain the square wave
gets spikes on it and as gain is pushed farther the whole shape
becomes ragged. Next, on the
you will be able to repeat the experiments you just conducted, using
the other two functions.
Now let's look at the iteration of the logistic function
in a different way. We have seen that under iteration, y and
therefore x may settle down to some value, independent of the
starting x. In successive iterations the y value gets closer and
closer to the limiting value. We saw this same behavior in some
of the "next 'x' selection" schemes we tried
earlier. Sometimes the function edges closer to the final value
from one direction, sometimes it closes in on the final value by
jumping back and forth over it. We call the final value a
A limit in mathematics is a number which some variable
approaches as time goes on. In this program you will be able to
observe that, as you iterate the chosen function, the value of y
sometimes gets closer and closer to a particular value. This is
the settling down behavior mentioned above. In a sense what we
will do on the Attractor Demonstration display is to test for the
presence of a limit or limits under the process of iteration. You
can see graphically the approach to a limit as you step through
Because with iteration the y value of a function sometimes
settles down at a particular level, it seems like there is
something about that value that attracts the function. Just as a
marble placed in a bowl will eventually settle in the bottom,
attracted by gravity to the lowest point. This idea of an
"attractor" for a mathematical function or a physical
phenomena, is a powerful one. We will see this concept in several
applications. As we will see, a minor change in our function
greatly affects the nature of an attractor. It may have a single
value or multiple values.
This idea of attractors applies
not only to the simple functions that we use for
demonstration but also to the functions that describe
complicated physical systems. For those functions the attractors
are exotic shapes in space not made up of the ordinary dimensions
which we may experience; dimensions of height, width and depth.
Those attractors are sometimes called strange attractors. The
space in which they exist is called phase space and may have many
dimensions of various measure depending on the dynamics of the
system being depicted. Run the
Attractor Demonstration display.
One value of the type display we have here is its intuitive
connection to the geometry of iteration. We see the iterations
bouncing between the curve of the function and the line y=x. The
nature of the attractor at any particular gain is related to the
geometry of these two lines. On the
display, look at our quadratic with
gains less than 1.
Now consider the situation for values of gain between 1 and 2.
In this region y is greater than the corresponding x up to the
point where the function curve intersects the y=x line. Look next
with gains less than 2.
look at gains
between 2 and 3. For gains between 2 and 3 then we have argued that the
attractor is the single value found at the intersection of the
function curve and the y=x line. Just as was the case for gains
between 1 and 2.
Above a gain of 3
, we see a different sort of attractor.
As we have been exploring the
behavior of the logistic function under iteration at various gain
settings we have made a point of looking at the attractor as it
split into 2, 4, 8 etc. values. You may have observed that the
change in gain required to get the next split decreases sharply
as the splits multiply. In fact each increment in gain is less
than 1/4 the previous increment. This means that there is room
for more than an infinite number of splits between a gain of 3
and a gain of 3.6. So what happens at values of
gain greater than 3.6?
The sort of fractured attractor we described in the last
display has been called chaotic. Those regions of gain that give
rise to attractors like this are called regions of chaos. Truly
the appearance is that of chaos. There seems to be no pattern to
the way the y value jumps around. Knowing where the current y is
does not allow you to easily predict where the next iteration
will take you. Keep in mind though that the next y is absolutely
determined by the geometry of the function, just as was the case
with the simpler attractors.
At a gain of 3.58 you will see that no matter how long you
iterate, certain y values never come up. There will be two broad
bands where the values lie, separated by a gap. The function
visits each of these bands in turn as was the case when there was
a twofold attractor. In that case each part of the attractor was
single valued with the function alternating periodically between
them. In this case the attractor is still periodic in the sense
that the function alternates between branches, but each branch
holds infinitely many values.
Having explored the logistic function under iteration, it is
useful to look at the other two of our examples. The last display
in this section is a
where you have a lot of flexibility to play around with the
graphical iteration of the three functions we have defined. Try various gains, base iterations and numbers of iterations for each function.
Are there any questions?