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