

Now that's strange...
Now what happens if we change the parameters of amplitude of
the forcing function and/or the rate of energy loss. This in
effect gives us a whole new dynamical system that must be
explored. So we distinguish between changing the system and
changing the initial conditions of the system. The next display
is for a system that has quite a different sort of attractor.
For the chosen values of amplitude and drag there is a single
attractor which is a "chaotic", or a
"strange" attractor.
Notice the
irregularity of the Position/Velocity vs. Time view in the next
display. This fellow spends some time on the plus side of the
origin and some on the minus, and remains in the general
vicinity, so he meets our definition of an oscillator. However,
as you scan out along the time axis by repeated clicks on Action,
you will not find any true periodicity. In fact, pick any state,
a set of values for x and x', and that exact state will not
be repeated, if you watched forever. Run the
DMO Chaotic Attractor
display, P and V vs. Time view and come on back here so we can
talk about it.



The kind of unrepeatability you saw in this system is a tough
thing for those of us in the future predicting business. When a
system breaks into this chaotic behavior we are in big trouble as
far as saying in detail what the future state will be even in the
short term. Before we go on to look at the other views of this
system, run the
DMO Sensitivity to Initial Conditions
display.
The significance of
sensitive dependence on initial conditions is that the divergence
in system behavior grows exponentially in time. This means that
for a dynamical system whose attractor is chaotic, if the
conditions are disturbed by the slightest amount, the behavior
from that point on will begin to diverge such that after a time
it will have no relationship to what it would have been in the
absence of that disturbance.
With regard to the world's weather, this idea of sensitive
dependence to initial conditions has been called the butterfly
effect. The premise is that the beating of a butterfly's
wings in Canada might result later in a typhoon in the
Philippines. This of course supposes that the world's weather
as a dynamical system is operating in a mode where it is on a
chaotic attractor. That remains to be conclusively demonstrated
but it sure looks chaotic. What if the world itself as a
dynamical system has a chaotic history? It does not bode well for
time travel.

Given the wild behavior we have seen, is there anything we can
say about the future of a dynamical system in chaos. Well, of
course there is or we would not be wasting our time with it.
Let's go back to the
DMO Chaotic Attractor display and look at the Phase Space Projection view. Let it run a few seconds just to a sense of the shape of
the projection on the (x,x') plane. Then we can talk about
it.



The key to analyzing systems in
chaos is to remember that a chaotic attractor is still an
attractor. When we disturbed dynamical systems that were on
their attractor, be it a point attractor or a periodic attractor,
the state point in phase space temporarily wandered off but after
the transient died away they returned to the attractor. When the
attractor is chaotic the same effect is seen. The difference is
that the instantaneous value of the state variables at any time
after the disturbance, bears no predictable resemblance to the
values that would have pertained in the absence of the
disturbance except that we know that they both will lie on the
attractor somewhere.
By observing
the Phase Space Projection view of the system attractor, we can
begin to make some predictions. At least in the time you let the
plot develop, there seems to be certain values of x and x'
which are excluded from the attractor. if you like,
go back
and let it run longer here, to see if after more time passes the overall dimensions of the plot change or if there are any wild excursions into previously
uncharted states.

You should be gaining some comfort at this point that at least
for this system, the attractor inhabits a welldefined region of
the phase space. Also you might be curious about the odd shape
the return point markers seen to be making. They are sort of
smeared up and down in the x'dimension on the plus side of
the xaxis. More about that later. Now take a look at the
Chaotic Attractor
display in the Phase
Space Orbit view. Yaw and pitch it around awhile to begin to get
some sense of the general shape. Remember that no two states are
the same in the whole bundle of orbit strands, no matter how long
you let it run.
The idea of confining a
line of infinite length in a finite volume and requiring that it
never cross itself might give us some trouble in visualizing the
situation. Hopefully the displays so far have helped. The phase
space orbit is like a skein of yarn where each strand is
infinitely fine. The attractor is the region of phase space where
these strands tend to accumulate. If I disturb the system in some
way, displacing a strand from the attractor, time will bring it
back to that certain region. Thus the term "attractor"
is justified.



When we were dealing with periodic attractors, the phase space
orbit was a few welldefined loops. Here the attractor instead of
being a finite number of loops as it was for the periodic
attractors, is an infinite bundle of loops in the vicinity of the
time axis circle. These loops are a hopeless tangle but we know
they never intersect each other. If they did the conditions at
the intersection would be identical so that future development
would also be identical to the past, making the system repeat
perhaps with a very large periodicity. In fact is was only
recently in the history of mathematics that people became aware
that there was a distinction between a periodic attractor with
very long period and a chaotic attractor.
What is not obvious
from the Phase Space Orbit view is that the bundle of loops in
the orbit has an intricate structure of its own. The region in
phase space occupied by the orbit is not just a badly formed
bagel as it appears from pitch and yaw at (90,0) or a doorknob,
as it appears from pitch and yaw at (0,0). If we slice through
the orbit at a series of (x,x') planes spaced around the time
axis circle, we will see that the cross sections have an
intricate shape. The Orbit Sections view is where that structure
becomes visible. Run
the model
in the Orbit Sections view after setting the pitch to 60 degrees and the yaw to 65
degrees to see this. Go ahead and do this now. Allow quite
a bit of time for this view to develop.

The Orbit Section view suppresses the orbit plot between sectioning planes
and illuminates the pixels where the orbit hits the planes in a
succession of colors so as to make the cross section details
visible. What we see now is that the attractor does not occupy
all the phase space included in the bundle of orbit loops. Rather
it apparently exists in convoluted layers. The Poincaré
Section view takes one of these cross sections and expands it for
a more detailed look at this internal structure. Run
the model
again in that view. Since we
have to integrate a full orbit loop to get a single pixel lit,
this view takes time to develop.
This is probably a good point to remind you that the fantastic
images we are looking at arise from the behavior of a real
mechanical system. These images take on a life of their own and I
tend to get fascinated by the images themselves. The object of
all this though is to be able to make at least some predictions
about the future of a real system.
The
structure revealed in the Poincaré section of our
attractor looks like the layers one might see in pastry dough
which had been repeatedly folded and rolled. In fact if we took a
bundle of orbit strands which initially ran parallel and
stretched the bundle out into a thin band, then folded the band
back on itself, repeating the process a few times we would find
the result would look much like the skein of strands we see in
our chaotic orbit. Strands initially close together would now be
far apart and vice versa. This stretching, folding and flattening
has a mixing effect that is the origin of the sensitivity to
initial conditions we observed earlier.
The Poincaré section of the chaotic attractor reveals
the details of the attractor's structure. You should have
noticed that there were areas where the orbit strands accumulated
and areas where the strands never went. This boundary between
light and dark in the Poincaré section is irregular in the
extreme. This sort of wild border region between certain domains
is a recurring theme in the mathematics of chaos. Examination of
highly irregular boundaries such as we see here is a matter of
great interest. So irregular is this boundary that any section of
the (x,x') plane that contains a strand of the orbit will
also contain part of the boundary, however small that section may
be.



The question, "How long is the coast of Maine?" is
not easily answered. If you measure the Maine coastline with a
yardstick you would miss a lot of tiny coves and inlets less than
three feet across. A onefoot ruler would measure more precisely
than a yardstick. A onecentimeter ruler more precisely than a
onefoot rule. The length of the Maine coast depends on the
length of your ruler and in the limit as the ruler length
approaches zero, the coast length becomes extremely long almost
infinite. Likewise the boundary of the Poincaré section of
the chaotic attractor we have plotted. The closer you look, the
more detail is revealed.
A slice through the chaotic attractor for Duffing's
mechanical oscillator then has a boundary that is infinite in
length but contained in a finite region of the (x,x') plane.
Regions of space which have an infinite boundary but finite
extent are called "fractals". We are accustomed to
think of a line as being of one dimension and a plane as being of
two dimensions. A fractal boundary is more than a line but less
than a plane and sure enough it has dimension between one and
two. A fractional dimension. Very strange. The fractional
dimension and fractured nature give rise to the name fractal.
Of course as we continue to zoom in, we still need to
calculate the orbit loops that never penetrate our section of
the (x,x') plane so the time to illuminate a certain number
of pixels in the image goes up dramatically as the size of the
viewing window decreases. This limitation is inherent in
mathematical modeling. If it were possible to get an analytical
expression for the solution to the model differential equation we
could jump in with any value of time and immediately calculate
the state. This sort of random access is one of the benefits of
the explicit solution.

The fact is with motions as complicated as we have uncovered
here, there is no solution available so we must go chronon by
chronon in a serial fashion to any point we want to see. It is
small wonder that this kind of analysis had to wait for the
computer to be invented. Forty years ago people would have had to
work for a lifetime to get 0.1% of the information you have seen
in the last few minutes.
Suppose now we had a stream of data from some dynamical system
which appeared to jump around erratically, without order. If we
could plot a Poincaré section of that data and it revealed
a cross section we could recognize as a fractal, we would know
that we were not dealing with random data but with a dynamical
system on a chaotic attractor. That would be a piece of future
prediction worth having.
The one remaining view is the Basins of Attraction. Since we
have only one attractor for this dynamical system, it is not of
much use here. You may
run
the view to
see what path any particular set of starting conditions takes to
reach the attractor if that is of interest to you. In the full powered (offline) version of this program, you may actually map the basins of attraction of the periodic attractors we studied earlier, as shown at the right.
The last lesson in this section on dynamical systems is a
research lab where you will be able to set the parameters that
define a Duffing mechanical oscillator and explore it for
attractors. I will provide several examples but after that you
are on your own.
Are there any questions?


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