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In this video, I'll say more about the
phase plane, and the types of behaviors
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that are possible for these types
of differential equations.
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So first, let me say just a little bit
about types of fixed points.
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As before we can have stable and
unstable fixed points.
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In 2D this comes with a twist,
almost literally.
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So this might be a stable fixed point, or
an attractor; it would have a bunch of
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phase lines, or trajectories
in the phase plane.
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I should mention, all of these are going
to be in some sort of a phase plane.
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So this would be an attracting fixed
point.
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Trajectories are drawn towards it,
getting closer and closer to this point.
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We could also have a situation though,
let's see if I can draw this,
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where trajectories spiral
in to a fixed point.
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Let me draw the fixed points are
the equilibrium points in red.
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These blue lines, they're getting pulled
in, it's stable.
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But there's a twist. One spirals in as one
approaches this point.
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Then there are two other varieties, you
can probably guess these.
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We can have a repelling equilibrium, it
might look something like this.
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If you're exactly on the red dot you stay
there but if you move off a little bit
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you get pushed away. So this is like
being at the top of a mountain.
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You move in any direction and away
you go.
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And then we can also have something like
this. So this is also unstable.
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If you fall off this mountain you get, you
don't return to the top of the mountain.
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So this is like being at the bottom of a
valley. Here you're at the top of a
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mountain but somehow if you fall off
that top you don't roll down the side,
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you spiral away from the mountain.
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So these are repellers, these are
attractors.
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These are sometimes called sinks and these
sources; this terminology is a little more
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common looking at two dimensional
differential equations.
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One might call this a spiral sink and
this a spiral source.
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"Source" because it's a source of lines,
lines are coming out of it.
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In any event, these are the four different
types fixed points you can have for a
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differential equation of this form.
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What about the Lotka-Volterra equation?
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We saw there that populations behaved
cyclically, cycled around.
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Are these cycles stable or unstable or
something else?
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So we'll figure this out experimentally.
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As usual, if we want to test something's
stability, we try iterating, or in this
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case solving, the differential equation
for slightly different initial conditions.
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When one does that on a phase plane like
this, one often ends up with what's
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called a phase portrait.
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Perhaps like a portrait in art, it
summarizes or highlights the key
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qualitative features of a system.
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So here's the Lotka-Volterra equations.
The version we looked at before.
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And we saw that for a certain initial
condition, I actually forget what it was,
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I think 10 foxes, 6 rabbits maybe?
We ended up with a cycle like this.
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So what happens if we try some different
initial conditions?
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So I can do that for each one. Each
initial condition I would get a rabbit
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curve and a fox curve and then I could
plot them together like I did here.
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If I did that we'd end up with this.
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Let me put some arrows on this. All these
curves are going in the same direction.
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So this outer curve is the one we had
before. Around and around it goes.
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With a different initial condition we get
a different cycle.
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And a different initial condition we get
a different cycle still.
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So each of these cycles, it turns out,
is neutral.
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What that means is that if we're in the
cycle and I kill a couple rabbits, that
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sounds like a terrible thing to do.
I don't kill a couple rabbits.
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I bring a couple more rabbits into
the system. Because I like rabbits.
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That would move us say
from here out to here.
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If a couple foxes got sick that
would move us down this way.
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The point is, if you're on this cycle, or
let's say if you're on this cycle,
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a small change in the number of rabbits or
foxes you'll end up at a different cycle.
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A cycle that's nearby. You don't get
pushed away, the rabbits and foxes
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don't take over the world. But you also
don't get pulled back to this cycle.
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So this tells you that depending on the
number of rabbits and foxes you have
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you'll be in one of these cycles.
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It turns out that there's a neutral fixed
point, an equilibrium point here in the
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middle. You can solve that using algebra.
That'll be one of the intermediate
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or advanced homework problems.
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So the main point is that this is an
example of a phase portrait.
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It shows a number of different
trajectories on the phase plane all at
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once and it let's us summarize the
behavior of a system.
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So we can see that there'll be cycles and
the cycles are neutral because the cycles
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don't get pushed away or pulled into
each other.
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So that's the phase portrait for the
Lotka-Volterra equations.
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There's one more example I'd like to
discuss because it will introduce us
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to a new type of behavior.
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It's a system of equations known as
the Van der Pol equations.
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Here they are. dX/dt=Y. dY/dt=-X+(1-X^2)Y.
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It's a two dimensional differential
equation like we've been studying before.
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One can solve this using Euler's method or
something similar and get solution curves.
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So let's look at some of
these solution curves.
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Here are X(t) and Y(t) for one set of
initial conditions. It looks like I
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chose an initial X=3 and an initial Y=-3.
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Y does this funny bend but pretty quickly
it gets into some regular cycle.
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Regular in that it repeats. It looks like
every 6 minutes, or whatever these units
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are, it's not exactly a sine wave
but whatever the shape is it repeats.
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Similar thing over here. These look like
shark fins or something.
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A little blip here but then very
quickly it settles into this cycle.
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Again, periodic it looks like every 6
minutes it does the same thing.
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So we can plot these two curves on a phase
plane and see what it looks like.
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We're going to get some sort of cycle,
a loop again.
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And here it is. Let me put some arrows on.
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So we started at X=3, Y=-3 and the
trajectory on the phase plane gets pulled
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into this cycle and then it loops around.
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And this isn't a perfect oval it's this
sort of almost trapezoidal sort of thing,
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parallelogram, I don't know what it is.
It's this. It's this little blob.
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And the system in X and Y cycle around.
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Ok, this isn't really anything new since
we've seen cycles for the Lotka-Volterra
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equation. But let's ask about their
stability.
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If I try a different initial condition
what do I get?
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So, I can do that. Here are results for a
different initial condition.
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X(t) and Y(t). I think this time I chose
0.2 and 0.2 for both X and Y.
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A little blip but then we see those shark
fins over here on the X curve.
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And on the Y curve we see whatever this
shape is, it looks a little like a tangent
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function, who knows what it is. It's this.
It's a thing with a little bend in it
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and then it repeats.
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So again, a little bit of transitory
behavior here but then it cycles into
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this period behavior.
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And let's take a look at that. We can plot
X vs. Y, so get rid of time on the phase
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plane and we see that this is the shape.
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So we start here, (0.2, 0.2), and we cycle
around. We do one little cycle and then
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we're back to this shape. So this blobby,
parallelogramy thing is an attractor.
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Why? Because multiple orbits get pulled
towards it.
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The phase portrait for this would show
several different initial conditions
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all getting pulled towars this.
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So let me show you what that looks like.
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Here it is. Again I'll draw on
a few arrows.
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There are two different initial
conditions.
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There's an initial condition here.
I might've chosen a slightly different
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initial condition. This looks like it's
even closer to the origin.
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I think I wanted to get an additional
loop in there.
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So if I start with an initial condition
very close to the origin it spirals out
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and approaches this sideways parallelogram
looking thing.
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If I start over here. This time I started
with +3 for X and Y. Again I get pulled
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in very quickly to this shape.
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So this shape is an attractor.
We have an attracting cycle.
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If I plotted additional initial conditions
that one gets pulled into the cycle,
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this one gets pulled into the cycle.
It's always the same cycle, this
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sideways blob looking thing.
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So the point is that this is an attractor,
and it's a periodic attractor.
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Cycles around and you get pulled into it
from the inside or from the outside.
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And this is sometimes
called a "limit cycle."
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I guess because in the limit of a long
time, everything tends to cycle in
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the same way.
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Perhaps it's interesting because this
isn't a perfect sinusoidal oscillation.
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It's not a circle or an oval. It's some
funny sort of non-linear type of thing.
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The key result here is that we've seen a
new type of behavior.
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In terms of these differential equations
we're seeing cycles and cycles that are
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attracting sometimes known as
a limit cycle.
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So far we've seen that two dimensional
differential equations of this form can
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have fixed points, equilibria that can be
attracting or repelling, and an orbit
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can fly off to infinity. I haven't shown
an example of that but it's not hard to
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construct one.
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We can have cycles; neutral cycles like in
the Lotka-Volterra equation or an
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attracting cycle, sometimes called a
limit cycle, in the Van der Pol equations
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that we just saw.
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But what about chaos? Can we have chaos
in these two dimensional systems?
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Let's think about this. I'll draw a phase
plane again.
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Let's remind ourselves of what chaos is.
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Chaos is a deterministic dynamical system;
yep, we've got determinism.
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The orbits need to be bounded, aperiodic,
and have sensitive dependence on
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initial conditions.
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Let's focus on those middle two criteria.
An orbit that's bounded and is aperiodic.
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So let's say that we have some region
that bounds the orbit. Maybe, arbitrarily
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I'll put a bound here in purple. Let's say
we want our orbit to stay within that
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region. Choosing it arbitrarily, just for
the sake of argument.
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So, can we have an aperiodic phase
line, aperiodic trajectory in here?
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Well, you can already see that's going to
be pretty hard.
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So I want something that, let's see so it
can't ever cycle around because that would
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not be aperiodic that's periodic.
So we can't have that.
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Now I've messed up my graph. Let's ignore
that and choose a different color.
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I'll do red.
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So now I want to wander around here and
I can never cross the line. I can't ever
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intersect this line. If I did intersect,
like I did here in blue, then because of
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determinism I have to repeat. If you get
to this point, move around, and you get
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to that point again, you have to do the
same thing. That's what determinism means.
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I'm in this bounded region and I can't
ever cross and so imagine this trajectory
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moving around, who knows what this would
look like in terms of X(t) and Y(t).
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What I'm trying to illustrate here is that
as I draw this line I'm making it harder
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and harder for this line to keep moving
around without bumping into it.
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If it bumps into it, game over, it's
periodic, it can no longer be aperiodic.
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So the fact that we're never allowed to
cross the line because of determinism,
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as we discussed in the previous video,
limits the behavior that's possible.
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A way I like to think of this is imagine
you're painting a floor and you're
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painting a floor red for some reason.
I don't know why you'd paint a floor red
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but you're painting a floor red so here
you are painting but you can't cross the
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line that you paint so if you do it wrong
you're boxing yourself into a corner and
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you have less and less room to maneuver.
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In any event, the question is "Is chaos
possible?" and the answer to this
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question is "no." And this was proved, I
think around 1900 by PoincarĂ© and then
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the theorem was strengthened a little bit
by Bendixson.
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So this is known as the PoincarĂ©-
Bendixson theorem.
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Let me see if I can spell that.
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Which says bounded, aperiodic behavior
is not possible for 2D differential
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equations of this form.
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So I hope this seems almost intuitive.
That the restriction that we can only
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draw in the box and we can never cross
a line means that it's not possible for
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two dimensional differential equations
to show aperiodic behavior.
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It's not immediately obvious, maybe that's
why there's something to prove here and
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it's a theorem and not a totally obvious
thing. I suppose one could imagine a curve
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that snakes in some incredibly intricate
fractal way but presumably those are
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eliminated mathematically if we restrict
ourselves to this sort of thing.
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So the bottom line is we can't see chaotic
behavior in two dimensional differential
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equations. They're more interesting than
one dimensional differential equations
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because we can see cycles, neutral cycles
and attracting cycles and these funny
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non-linear cycles like we saw in the
Van der Pol equations,
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but chaos is not possible.
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For a differential equation to show
chaotic behavior we'll have to be in
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dimensions higher than two.
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We'll talk about that in a subsequent sub-
unit of this section.