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Continuing with our review,
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the next topics that we covered concern
bifurcation diagrams.
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They're a way to see how the behavior of
a dynamical system changes as a parameter
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is changed.
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00:00:15,270 --> 00:00:20,220
I think it's best to think of them as
being built up one parameter value
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at a time.
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So, for each parameter value, make a phase
line if it's a differential equation,
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or a final-state diagram for an
iterated function.
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And you get a collection of these, and
then you glue these together to make a
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00:00:32,120 --> 00:00:34,550
bifurcation diagram.
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So, here's one of the first bifurcation
diagrams we looked at.
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This is the logistic equation with
harvest. The equation is down here.
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And so, h is the parameter that
I'm changing.
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H is here, it goes from 0 to 100 to 200
and so on.
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And so, a way to interpret this is
suppose you want to know what's going on
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at h is 100. Well, I would try to focus
right on that value, and I can see "aha,"
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it looks to me like there is an attracting
fixed point here, and a repelling
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fixed point there. So there are two fixed
points: one of them attracting and one of
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them repelling, or repulsive.
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And, what's interesting about this is that
so this is is the stable fixed point,
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this would be the stable population of
the story I told involved fish in a lake
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or an ocean, and h is the fishing rate,
how many fish you catch every year.
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And that increases, and as you increase h,
the sort of steady state population of the
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00:01:41,619 --> 00:01:43,879
fish decreases, that makes sense.
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But what's surprising is that when
you're here, and you make a tiny increase
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in the fishing rate, the steady-state
population crashes and in fact disappears.
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The population crashes.
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So, you have a small change in h leading
to a very large qualitative change in the
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00:02:03,437 --> 00:02:04,657
fish behavior.
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So, this is an example of a bifurcation
that occurs right here.
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00:02:08,007 --> 00:02:12,007
It's a sudden qualitative change in the
system's behavior as a parameter
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00:02:12,007 --> 00:02:15,647
is varied slowly and continuously.
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00:02:15,647 --> 00:02:19,647
So, we looked at bifurcation diagrams for
differential equations and we saw the
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00:02:19,647 --> 00:02:25,187
surprising discontinuous behavior, then we
looked at bifurcation diagrams for the
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00:02:25,187 --> 00:02:30,837
logistic equation, and we saw bifurcations
here is period 2 to period 4, but what was
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00:02:30,837 --> 00:02:34,837
really interesting about this was that
there's this incredible structure to this
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00:02:34,837 --> 00:02:37,077
and we zoomed in and it looked really
cool.
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00:02:37,077 --> 00:02:41,077
There are period 3 windows, all sorts of
complicated behavior in here.
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00:02:41,077 --> 00:02:45,077
So, there are many values for which the
system is chaotic.
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00:02:45,077 --> 00:02:49,077
The system goes from different period
to period in a certain way.
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00:02:49,077 --> 00:02:54,817
And this has a self-similar structure:
it's very complicated but there is some
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00:02:54,817 --> 00:03:01,367
regularity to this set of behavior for the
logistic equation.
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00:03:01,367 --> 00:03:06,117
So, then we looked at the period doubling
route to chaos a little bit more closely.
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00:03:06,117 --> 00:03:09,127
And in particular I defined this ratio,
delta.
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00:03:09,127 --> 00:03:13,127
It tells us how many times larger branch n
is than branch n+1.
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00:03:13,127 --> 00:03:18,927
So, delta is how much larger or longer
this is than that.
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00:03:18,927 --> 00:03:22,317
That would be delta 1. How much longer,
how many times longer is this length
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00:03:22,317 --> 00:03:26,617
than that?
That would be delta 2.
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00:03:26,617 --> 00:03:31,687
And we looked at the bifurcation diagrams
for some different functions,
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00:03:31,687 --> 00:03:39,397
and I didn't prove it, but we discussed
how this quantity, delta, this ratio
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00:03:39,397 --> 00:03:43,237
of these lengths in the bifurcation
diagram is universal.
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00:03:43,237 --> 00:03:46,237
And that means it has the same value for
all functions
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00:03:46,237 --> 00:03:49,707
provided, a little bit of fine print,
they map an interval to itself and have a
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00:03:49,707 --> 00:03:51,627
single quadratic maximum.
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00:03:51,627 --> 00:03:56,347
So, this value, which is I believe
known to be a rational and I think
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00:03:56,347 --> 00:04:02,837
transcendental, is known as Feigenbaum's
constant, after one of the people who made
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00:04:02,837 --> 00:04:05,187
this discovery of universality.
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This is an amazing mathematical fact
and points to some similarities among
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00:04:10,257 --> 00:04:13,157
a broad class of mathematical systems.
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00:04:13,157 --> 00:04:17,157
To me, what's even more amazing is that
this has physical consequences.
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00:04:17,157 --> 00:04:21,017
Physical systems show the same
universality.
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00:04:21,017 --> 00:04:24,577
So, the period doubling route to chaos
is observed in physical systems.
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00:04:24,577 --> 00:04:28,577
I talked about a dripping faucet and
convection rolls in fluid, and one can
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00:04:28,577 --> 00:04:31,287
measure delta for these systems.
It's not an easy experiment to do,
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00:04:31,287 --> 00:04:32,767
but it can be done.
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00:04:32,767 --> 00:04:37,777
And the results are consistent with this
universal value, 4.669.
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00:04:37,777 --> 00:04:43,407
And so, what this tells us is that somehow
these simple one-dimensional equations,
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00:04:43,407 --> 00:04:46,067
we started with a logistic equation,
an obviously made-up story about
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00:04:46,067 --> 00:04:51,357
rabbits on an island, that nevertheless
produces a number, a prediction
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00:04:51,357 --> 00:04:55,357
that you can go out in the real physical
world and conduct an experiment
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00:04:55,357 --> 00:04:59,357
with something much more complicated
and get that same number.
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So, this is I think one of the most
surprising and interesting results
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in dynamical systems.
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00:05:08,297 --> 00:05:12,087
So, then we moved from one-dimensional
differential equations to two-dimensional
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differential equations.
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00:05:13,767 --> 00:05:17,337
So now, rather than just keeping track
of temperature or population, we're
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00:05:17,337 --> 00:05:22,607
going to keep track of two populations,
say R for rabbits and F for foxes.
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And we would have now a system of two
coupled differential equations:
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00:05:26,607 --> 00:05:30,357
the fate of the rabbits depends on
rabbits and foxes, and the fate of the
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00:05:30,357 --> 00:05:33,477
foxes depends on foxes and rabbits.
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00:05:33,477 --> 00:05:35,827
So, they're coupled, they're
linked together.
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And one can solve these using Euler's
method or things like it,
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very, almost identically to how one would
for one-dimensional differential equations
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And you get two solutions:
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00:05:45,837 --> 00:05:47,657
you get a rabbit solution and a fox
solution.
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And in this case, this is the
Lotka-Volterra equation,
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00:05:51,657 --> 00:05:55,657
they both oscillate.
We have cycles in both rabbit and foxes.
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But then, we could plot R against F.
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So, we lose time information, but it will
show us how the rabbits and the foxes
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00:06:03,657 --> 00:06:05,837
are related.
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00:06:05,837 --> 00:06:09,837
And if we do that, we get a picture that
looks like this.
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Just a reminder that this curve goes in
this direction.
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00:06:13,837 --> 00:06:19,747
And so, the foxes and rabbits are
cycling around.
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00:06:19,747 --> 00:06:23,747
The rabbit population increases,
then the fox population increases.
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00:06:23,747 --> 00:06:27,747
Rabbits decrease because the foxes
are eating them.
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00:06:27,747 --> 00:06:31,107
Then the foxes decrease because
they're sad and hungry because
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there aren't rabbits around, and so on.
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So, this is is similar to the phase line
for one-dimensional equations,
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but it's called a phase plane because it
lives on a plane.
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00:06:40,307 --> 00:06:43,337
And this hows how R and F are related.
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Phase plane and then phase space is
one of the key geometric constructions,
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analytical tools used to visualize
behavior of dynamical systems.
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So, an important result is that there
can be no chaos, no aperiodic
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solutions in 2D differential equations.
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So, curves cannot cross in phase space.
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The equations are deterministic, and that
means that every point in space, and
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00:07:12,317 --> 00:07:16,317
remember this is in phase space, so my
point in space gives the rabbit and fox
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population, there's a unique direction
associated with the motion.
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DF/DT, DR/DT, that gives a direction.
It tells you how the rabbits are
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increasing, how the foxes are increasing.
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If two phase lines ever cross, like they
do where my knuckles are meeting,
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then that would be a non-deterministic
dynamical system.
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There would be two possible trajectories
coming from one point.
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So, the fact that two curves can't cross
in these systems limits the behavior.
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They sort of literally paint themselves in
as they're tracing something out, tracing
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a curve out in phase space.
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So there can be stable and unstable fixed
points and orbits can tend toward infinity
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00:08:01,147 --> 00:08:04,927
of course, and there can also be limit
cycles attracting cyclic behavior, and we
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00:08:04,927 --> 00:08:06,537
saw an example of that.
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But the main thing is that there can't be
aperiodic orbits.
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And that result is known as the
Poincaré-Bendixson theorem.
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It's about a century old.
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00:08:15,807 --> 00:08:19,807
And it's not immediately obvious;
it takes some proof.
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Like I said, that's maybe why it's a
theorem and not just an obvious statement.
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One could imagine, and people in the
forums have been trying to imagine
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space-filling curves that somehow never
repeat but also never leave a bounded area
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But the Poincaré-Bendixson theorem says
that those solutions somehow
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aren't possible.
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So, the main result is that
two-dimensional differential equations
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cannot be chaotic.
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That's not the case for three-dimensional
differential equations, however.
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So, here are the Lorenz equations.
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Now, again it's a dynamical system, it's
a rule that tells how something changes
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in time.
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00:09:01,247 --> 00:09:08,167
Here that something is x, y, and z, and
I forget what parameter values I chose
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00:09:08,167 --> 00:09:09,597
for sigma, rho, and beta.
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00:09:09,597 --> 00:09:12,687
And we can get three solutions:
x, y, and z.
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And these are all curves plotted as a
function of time.
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But we could plot these in phase space,
x, y, and z together.
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And for that system, if we do that, we get
some complicated structure that
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loops around itself and repeats.
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It looks like the lines cross, but
they don't.
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There's actually a space between them.
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It looks like they cross because this is
a two-dimensional surface trying to
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00:09:36,247 --> 00:09:39,697
plot something in 3D.
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00:09:39,697 --> 00:09:46,587
Alright, so just a little bit more about
phase space.
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Determinism means that curves in phase
space cannot intersect.
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But because the space is three-dimensional
curves can go over or under each other.
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And that means that there is a lot more
interesting behavior that's possible.
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00:09:58,577 --> 00:10:02,577
A trajectory can weave around and under
and through itself
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00:10:02,577 --> 00:10:04,557
in some very complicated ways.
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00:10:04,557 --> 00:10:08,057
What that means, in turn, is that
three-dimensional differential equations
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00:10:08,057 --> 00:10:09,367
can be chaotic.
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00:10:09,367 --> 00:10:16,047
You can get bounded, aperiodic orbits,
and it has sensitive dependence as well.
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00:10:16,047 --> 00:10:19,367
And then we saw that chaotic trajectories
in phase space are particularly
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00:10:19,367 --> 00:10:20,847
interesting and fun.
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They often get pulled to these things
called strange attractors.
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00:10:24,247 --> 00:10:30,457
So here's the Lorenz attractor or the
famous values for the Lorenz equation.
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00:10:30,457 --> 00:10:32,737
Strange attractors.
What are strange attractors?
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00:10:32,737 --> 00:10:36,737
Well, they're attractors, and what that
means is that nearby orbits get
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00:10:36,737 --> 00:10:38,017
pulled into it.
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00:10:38,017 --> 00:10:41,537
So, if I have a lot of initial conditions,
they all are going to get
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pulled onto that attractor.
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So, in that sense, it's stable.
If you're on that attractor and somebody
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00:10:47,177 --> 00:10:50,337
bumps you off a little bit, you'd get
pulled right back towards it.
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00:10:50,337 --> 00:10:52,057
That's what it means to be stable.
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00:10:52,057 --> 00:10:54,797
So, it's a stable structure
in phase space.
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00:10:54,797 --> 00:10:58,797
But the motion on the attractor is not
periodic the way most attractors that
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00:10:58,797 --> 00:11:01,027
we've seen are, or even fixed points.
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But the motion on the attractor
is chaotic.
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00:11:03,697 --> 00:11:07,047
So, once you're on the attractor, orbits
are aperiodic and have
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00:11:07,047 --> 00:11:09,387
sensitive dependence on
initial conditions.
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00:11:09,387 --> 00:11:12,227
So, it's an attracting chaotic attractor.
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00:11:16,043 --> 00:11:20,043
Then we looked at this a little bit more
geometrically and I argued that the key
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00:11:20,043 --> 00:11:23,603
ingredients to make a strange attractor
or to make chaos of any sort, actually,
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00:11:23,603 --> 00:11:25,243
is stretching and folding.
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00:11:25,243 --> 00:11:28,833
So, you need some stretching to pull
nearby orbits apart.
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00:11:28,833 --> 00:11:32,833
The analogy I discussed was
kneading dough.
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00:11:32,833 --> 00:11:35,613
So, when you knead dough, you stretch it.
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00:11:35,613 --> 00:11:39,613
That pulls things apart, and then you fold
it back on itself.
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00:11:39,613 --> 00:11:41,473
So, the folding keeps orbits bounded.
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00:11:41,473 --> 00:11:44,903
It takes far apart orbits and moves them
closer together.
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00:11:44,903 --> 00:11:47,473
But stretching pulls nearby orbits apart,
and that's what leads to
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00:11:47,473 --> 00:11:50,683
the butterfly effect,
or sensitive dependence.
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00:11:50,683 --> 00:11:55,433
Now, stretching and folding, it may be
relatively easy to picture in
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00:11:55,433 --> 00:11:59,433
three-dimensional space, either a space
of actual dough on a bread board
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00:11:59,433 --> 00:12:01,143
or a phase space.
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00:12:01,143 --> 00:12:04,653
But it occurs in one-dimensional maps
as well, the logistic equation stretches
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00:12:04,653 --> 00:12:06,103
and folds.
190
00:12:06,103 --> 00:12:10,103
And this can explain how one-dimensional
maps, iterated functions, can capture
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00:12:10,103 --> 00:12:14,103
some of the features of these
higher dimensional systems.
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00:12:14,103 --> 00:12:21,723
And it begins to explain, also, how these
higher dimensional systems,
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00:12:21,723 --> 00:12:26,733
convection rolls, dripping faucets,
can be captured by one-dimensional
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00:12:26,733 --> 00:12:31,563
functions like the logistic equation and
this universal parameter, 4.669.
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00:12:31,563 --> 00:12:36,263
So, in any event, stretching and folding
are the key ingredients for a chaotic
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00:12:36,263 --> 00:12:38,473
dynamical system.
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00:12:38,473 --> 00:12:43,533
So, strange attractors once more.
They're these complex structures that
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00:12:43,533 --> 00:12:45,783
arise from simple dynamical systems.
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00:12:45,783 --> 00:12:48,583
A reminder that we looked at three
examples: the Hénon map, the Hénon
200
00:12:48,583 --> 00:12:54,473
attractor, which is a two-dimensional
discrete, iterated function.
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00:12:54,473 --> 00:13:00,093
And then, two different sets of coupled
differential equations in three dimensions
202
00:13:00,093 --> 00:13:04,483
the famous Lorenz equations, and also
the slightly less famous but equally
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00:13:04,483 --> 00:13:06,883
beautiful Rössler equations.
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00:13:06,883 --> 00:13:10,643
Again, the motion on the attractor is
chaotic, but all orbits get pulled
205
00:13:10,643 --> 00:13:11,623
to the attractor.
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00:13:11,623 --> 00:13:15,623
So, strange attractors combine elements
of order and disorder.
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00:13:15,623 --> 00:13:18,163
That's one of the key themes
of the course.
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00:13:18,163 --> 00:13:20,883
The motion on the attractor is locally
unstable.
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00:13:20,883 --> 00:13:24,883
Nearby orbits are getting pulled apart,
but globally it's stable.
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00:13:24,883 --> 00:13:31,883
One has these stable structures, the same
Lorenz attractor appears all the time.
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00:13:31,883 --> 00:13:35,383
If you're on the attractor, you get pushed
off it, you get pulled right back in.
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00:13:37,816 --> 00:13:41,816
Alright, and the last topic we covered in
unit 9 was pattern formation.
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00:13:41,816 --> 00:13:45,056
So, we've seen throughout the course in
the first 8 units that dynamical systems
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00:13:45,056 --> 00:13:46,886
are capable of chaos.
215
00:13:46,886 --> 00:13:49,106
That was one of the main results.
216
00:13:49,106 --> 00:13:51,296
Unpredictable, aperiodic behavior.
217
00:13:51,296 --> 00:13:54,246
But there's a lot more to dynamical
systems than chaos.
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00:13:54,246 --> 00:13:58,246
They can produce patterns, structure,
organization, complexity, and so on.
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00:13:58,246 --> 00:14:01,096
And we looked at just one example
of a pattern-forming system.
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00:14:01,096 --> 00:14:03,276
There are many, many ones to choose from.
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00:14:03,276 --> 00:14:05,686
But we looked at reaction-diffusion
systems.
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00:14:07,491 --> 00:14:10,301
So there, we have two chemicals that
react and diffuse.
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00:14:10,687 --> 00:14:15,737
And diffusion, that's just the random
spreading out of molecules in space,
224
00:14:15,737 --> 00:14:19,617
diffusion tends to smooth out
differences, it makes everything
225
00:14:19,617 --> 00:14:22,317
as boring and bland as possible.
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But, if we have two different chemicals
that react in a certain way,
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it's possible to get stable spatial
structures even in the presence
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of diffusion.
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Here are these equations-- I described
them in the last unit.
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This is deterministic, just like the
dynamical systems we've studied before.
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And it's spatially extended, because now
U and V are functions, not just of T, but
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of X and Y.
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So, these become partial differential
equations.
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Crucially, the rule is local.
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So, the value of U or the value of V,
those are chemical concentrations,
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depend on some function of a current
value at that location, and on this
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Laplacian derivative at that location.
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So, we have a local rule in that the
chemical concentration here doesn't know
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directly what the chemical concentration
is here; it's just doing it's own thing
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at its own local location.
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Nevertheless, it produces these
large-scale structures.
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So, just one quick example.
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We experimented with reaction-diffusion
equations at the
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Experimentarium Digitale site.
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00:15:28,828 --> 00:15:31,948
Here's an example that we saw emerging
from random initial conditions,
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these stable spots appear.
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00:15:35,168 --> 00:15:41,828
And then we also looked at a video from
Stephen Morris at Toronto where two fluids
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are poured into this petri dish, and like
magic, these patterns start to emerge
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out of them.
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So, Belousov Zhabotinsky has another
example of a reaction-diffusion system.
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So, pattern formation is a giant subject.
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It could be probably a course in and of
itself.
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00:15:58,833 --> 00:16:01,703
The main point I want to make is that
there's more to dynamical systems
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than just chaos or unpredictability
or irregularity.
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00:16:04,963 --> 00:16:09,213
Simple, spatially-extended dynamical
systems with local rules are capable
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00:16:09,213 --> 00:16:13,213
of producing stable, global patterns and
structures.
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So, there's a lot more to the study of
chaos than chaos.
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Simple dynamical systems can produce
complexity and all sorts of interesting
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emergent structures and phenomena.