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 PoincarĂ© and then
the theorem was strengthened a little bit
by Bendixson.
So this is known as the PoincarĂ©-
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.