In this last segment about maps, I'm going
to talk about the universality of chaos,
beginning with the Feigenbaum number.
Recall this picture of the bifurcation diagram
of the logistic map. It's pretty clear that
the bifurcations in this cascade aren't
evenly spaced, rather, they are getting
smaller as r increases and as the period
increases with r. If you look closely, you'll
see that the width and height of those little
pitchforks are decreasing at a constant ratio;
that is, that the ratio of the width delta 3
to the width delta 2, is about the same
size as the ratio of the width delta 2 to
delta 1. In the high period limit, that ratio
approaches the value of about 4.66.
Mitchell Feigenbaum is responsible for the proof
about that ratio, and that number bears his name.
Now, you'll notice the limit in this equation,
what that means is that delta 1 over delta 2 is close
to 4.66, delta 2 over delta 3 is closer to
4.66, and so on and so forth. And in the
limit of very small bifurcations, the number
actually limits to the Feigenbaum number.
Note that the Feigenbaum number is about the
widths of the pitchforks. There is a similar
result, but with a different ratio for the
heights. Now, what's really amazing about
this result is that it holds for any 1-D function
with a quadratic maximum; that is, it goes
well beyond the logistic map. It's also true for
the sine map and it's also true for any other
map that looks like a parabola near its
maximum. In making the point about how
amazing this is, Steve Strogatz says the
Feigenbaum number is a new physical constant
as fundamental to 1-D maps as pi is to circles.
This is one manifestation of the universality
of chaos. Systems as diverse as hurricanes,
orbiting moons, pendulums, and the human heart
all act the same in the throws of chaos. This
is a large part of what fascinates me and
I think many other people about this field;
this universality. And the Feigenbaum number
is your first taste of that. Don't take this
number too far. Note: 1-D. Not 2-D or more.
Also, maps. Not flows. There is some evidence
that this result holds for some higher dimensions
and for some flows, but the proofs don't extend
to them, so it may well be true, but we don't
know for sure yet. Speaking of higher dimensional
maps, here is an example of a 2-dimensional
map. It's called Smale's Horseshoe. Smale's Horseshoe
operates not on the unit interval like the
logistic map, but on the unit square. And here's
the action of the map. The first thing it does
is stretch out the square to be a long,
thin rectangle with the same volume. The next
action of the map is to take that rectangle
and fold it and then chop off the extra bits.
So it's a map of the unit square onto itself
and it has the interesting property of taking
points that are far apart and bringing them
close together. At the same time, it takes
some points that are close together and
maps them very far apart. The stretching and
folding that creates this is a paradigm in
chaos. It is the cause of sensitive dependence
on initial conditions. It happens in the
logistic map too because that quadratic
transformation kneads the unit interval.
And when I say 'kneads' I don't mean that
kind of 'needs', I mean this kind of 'kneads',
like bread. And speaking of kneading and
bread, here is an interesting 3-dimensional
map. This is a lump of bread dough with a
little bit of dye put on it. As in the case
of Smale's Horseshoe, the kneading of
the bread causes points that are close
to end up far apart and far away points to end up close
together. Again, that's the source of sensitive
dependence on initial conditions. It's also
why bakers knead dough. This progression
of images shows a ball of dough, again
with a tiny bit of red dye in it, after a
succession of applications of the kneading
map. The kneading map is taking that ball
of dough, squishing it flat, folding it over
and gathering it into a ball again. As you
can see, working forward from the top left
image, which is the initial image before any
iterations of the kneading map were applied,
to the second image, the top middle image which
is after one application of the kneading map,
the dye gets distributed pretty quickly through
the bulk of the dough. Of course, thinking of
kneading bread as a map, a single discrete
operation, isn't really right. The action is
proceeding continuously in space and time as
our hands move the dough around continuously.
We can only think of this as a map, a
discrete time operation, if we kind of look
away during the kneading operation. If
we only look at the dough when it's back
in a ball after each operation,
that's kind of like shining a strobe light
at a physical system.
This is the driven damped pendulum that I built a while
ago. At this particular force and frequency, the dynamics
are chaotic - never quite repeating, sensitively
dependent on initial conditions, and yet, patterned.
If I turn off the lights and shine a strobe light
at the pendulum, you only observe it every
second or so. You have no idea what it did
in between. You're only information is at
those discrete intervals. And that's the difference
between a map and a flow - discrete time vs.
continuous time.
We've gone through a lot of the important
concepts in nonlinear dynamics in the context
of maps. In the next unit of this course,
we're going to circle around through a lot
of those same concepts in the context of
flows. So you can think of this like looking
at the full dynamics of the kneading operation
as you squish the dough, fold it, and gather it, rather
than just looking at it every time it's a ball
and then looking away while you do this,
and looking back.