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