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In the last subunit, we looked at the
Lotka-Volterra equations as our first
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example of a two dimensional
differential equation.
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In this subunit, I'd like to look at two
dimensional differential equations more
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generally and focus on properties
of the phase plane.
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And in this video, I'll start with
a couple of examples.
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So a reminder of what we're
working on here.
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We're looking at differential equations
of this form.
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So we have two variables, I'll call them
X and Y now instead of r and f for
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rabbits and foxes, any old variables.
And this is a dynamical system, it
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specifies how X and Y change but
because it's a differential equation it
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does so indirectly by telling us the rates
of change or the velocity of X and Y,
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not directly giving X and Y values.
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The rate of change of X is a
function of X and Y,
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the rate of change of Y is a
function of X and Y.
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These could be different functions and
note that X depends on Y in general,
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Y depends on X so we would say that these
two different differential equations are
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"coupled" because they
depend on each other.
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So one can solve these equations,
produce solutions using Euler's method
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or something like it.
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And then one gets two solution curves.
So let me show an example of that.
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Here are two possible solution curves
and if you did the quiz
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you've stared at these before.
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This is X of t, X(t), and this is Y of t,
Y(t), and they both wiggle and then they
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are approaching zero so it looks like
there's an attracting fixed point at zero.
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Let's think what the phase plane
might look like for this.
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So I'll try to do a rough sketch of this
and then I'll show you the plot
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that I had a computer do.
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OK, so I'll draw some axes first.
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So this is Y and this is X.
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And I just want to get a general picture
of the shape of this.
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So I start, initially X=-7 and Y=-3.
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So X is -7, Y is -3, that's going to put
me somewhere over here.
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That's my starting point.
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So -Y, -X.
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And I know that I'm going to end up here
and these wiggles indicate some kind of
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spiral and the big question is now which
direction does the spiral go?
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So let's see. So the first thing that
happens is Y increases and X is decreasing
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- X gets a little more negative
while Y increases.
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So that's going to end up looking
something like that.
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That sort of motion. So X is decreasing,
that's moving to the left,
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because X is going down here,
the value of X goes from -7 to -8.
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So X goes from -7 to -8, but Y is
increasing, from -3, -2, to all the way
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up here and it ends up at maybe +3.
That's going to end up around there.
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And then, it will spiral in like this.
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So this is a fixed point at 0, and it's
stable because points get pulled in and
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we have this sort of spiral thing.
We can't really spiral in quite the same
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way in one dimension,
but we can in two dimensions.
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So I think the best way to do this, to go
from these shapes to this shape
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is to think about the starting point,
think about the ending point and then
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what might happen in the middle.
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Another thing you could do, you could
sort of plot point by point.
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So t=5, we have an X of around 4, and a Y
of around -1.5. So maybe that's over here.
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So anyway this isn't designed to be
exactly to scale,
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but just give the general shape.
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Let me show you what a computer plot of
this would look like:
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Sneak that over here. Try to get this
all on the screen. There we go.
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And I should put arrows on this. My
program doesn't do that automatically.
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So we have something
spirally in to the origin.
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This is a stable fixed point at X=0, Y=0.
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OK, so that was one example.
Let me do one more.
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So here's another example.
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Suppose we have these two solution
curves; X is a function of t
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and Y is a function of t.
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In this case we have oscillatory behavior
but the amplitude doesn't decrease.
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So the amplitude and the
frequency are staying constant.
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So let's try to visualize this behavior,
oscillations in X, oscillations in Y.
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So if these were populations they'll
be perfectly cyclic.
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What would that look like
in the phase plane?
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So here are my axes, that is X against Y,
no time on the phase plane.
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And we start, it looks like I chose the
same starting point as before, Y=-3, X=-7.
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So X=-7, Y=-3. I'm going to start here.
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And then let's see,
X increases while Y decreases.
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Alright because I start here, X is going
up, that means I expect this blue thing
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to move to the right and Y is going down.
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So that motion is going to look
something like that.
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When X is 0, Y looks to be about -5.
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Then, Y starts increasing, that means I'm
going up in this direction,
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X is still increasing.
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We'll end up with motion
that looks like this.
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So X is going from roughly -10 to 10,
maybe that's 9.5, I don't know.
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These are sort of min and max values for X
Y is going roughly between 4.5 and -4.5.
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So this would be an ellipse.
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It's not a circle because this
is not the same as that.
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So the main point is that this type of
motion where we have two quantities
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that are oscillating sinusoidally on a
phase plane, will be some sort of
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an ellipse or oval.
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So this sort of motion means that X is
moving back and forth and
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Y is moving back and forth and
that those motions are in phase.
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Let me show you a computer
version of this plot.
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There it is. I'll put arrows on.
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So we can see X is going from a little bit
less than 9, a little bit more than -10
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between here, and the amplitude
of the Y is about 4.5.
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And so this would just cycle
around like this.
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So the main point of this is to --
Let's see, so we'll be describing
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motion in phase space like this
and in two dimensional phase space,
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the thing to bear in mind is that this
two dimensional phase space plot
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is just two solutions graphed
together without time.