So let's summarize what
we've done in Unit 7.

I introduced two and three
dimensional dynamical systems.

Two and three dimensional
differential equations

and two dimensional
iterated functions.

The main new thing or object to think
about from this unit was the idea of

phase space.

A phase plane or a phase space
or in the discrete dynamical system case

for the Hénon map, those xy-plots.

I started by introducing the
Lotka-Volterra equation.

This was the main example of a two
dimensional differential equation.

So we have two equations...uh oh, that
should be an R for rabbits.

Good thing I noticed that.

So we've got two differential equations
keeping track of two variables, R and F.

R, the rabbits, depend on the rabbits and
the foxes and the foxes depend on the

rabbits and the foxes.
So we'd say they're coupled.

The fates of the rabbits and
the foxes are intertwined.

These are known as Lotka-Volterra
equations and they are in ecology and

perhaps in economics as well.

Sort of the basic model, the most simple
model of a predator-prey interaction.

We can solve these differential equations
using Euler's method or the like and we

have two solution curves.

This is rabbits as a function of time and 
these are the foxes as a function of time.

We notice that they oscillate.

So we have these two solution curves.
To try to get a qualitative picture of

what's going on and how R and F, rabbits
and foxes, are related to each other

we can plot R against F
and get rid of time.

That's just a two dimensional version of
the phase lines we worked with before.

Doing so gives us a picture like this.

We plot R and F against each other and it
shows how R and F are related.

In this case we move
this way on the diagram.

The rabbit population increases then the
fox population increases while the rabbits

decrease and the foxes decrease and
then the rabbits come back again.

So we see how these
two cycles are related.

So that's the phase plane.

It's often useful to show several
different solutions on the same phase

plane. If there are multiple equilibria,
maybe show those. And that's sometimes

called a phase portrait. It gives an
overall view of the qualitative features

of the differential equations. By
qualitative I mean the long-term fate

of orbits. So this would be the phase
portrait for the Lotka-Volterra system.

We see that for different starting values
of rabbits and foxes we get different

cycles but in all cases we have cycles.
These cycles turn out to be neutral.

Meaning that they are neither attracting
nor repelling.

If you're on this cycle and move off a
little bit you're just on another cycle.

You don't get repelled away or pushed
back.

Another example that I considered was the
Van der Pol oscillator another two

dimensional differential equation.

This is interesting because it has what's
called a limit cycle.

The limit cycle itself is this repeating
shape here and we can see that multiple

orbits are drawn towards it.
So it's an attractor.

Here's an orbit from outside that gets
pulled in to the cycle.

Here's one from inside that gets pulled
in to the cycle.

In two dimensional differential equations
we can have oscillating solutions and we

can have cycles that are attractors; these
are known as limit cycles.

We talked some about of the consequences
of determinism in the phase plane.

So the dynamical system is deterministic.

It's just a rule that tells how R and F,
or x and y, whatever they are,

change in time.

Another way of saying that is that
the initial condition and the differential

equation, the rule, specifies a unique
path through the phase plane.

If I tell you the starting point and the
rule there's one and only one solution

to the equation.

As a result of this curves on
the phase plane cannot cross.

If they did cross it would violate
determinism. It would mean, that from

that point, there would be two possible
curves and that violates this notion

of determinism.

So curves, solution curves, on the phase
plane cannot cross each other and that

has some important consequences.

The fact that curves cannot cross limits
the possible long-term behavior of 2D

differential equations.

So what can those behaviors be? Well there
can be stable and unstable fixed points.

You could spiral in, in addition
to getting pulled in.

You could have a fixed point and
you spiral towards it.

Still fixed points, stable and unstable.

Orbits could tend to infinity.

There could be limit cycles like in the
Van der Pol oscillator which is a type

of attracting cyclic periodic behavior.

That's pretty much it. There's a famous
and important result known as the

Poincaré-Bendixson theorem. Which says,
simplifying somewhat, that bounded,

aperiodic orbits are not possible for two
dimensional differential equations.

Of course that assumes that the right hand
sides of the differential equations

aren't crazy and discontinuous or
something like that.

So the main consequence of this is that we
can't see chaotic behavior in two

dimensional differential
equations of this sort.

We can't see aperiodic, bounded orbits.

I should mention, by the way, that for
these 2D differential equations I've just

been presenting solutions from Euler's
method but there's some very nice

analytical techniques for 2D differential
equations but they're beyond the scope

of this course. They need calculus and
typically a little bit of linear algebra.

So in 1D we can characterize equilibria
points by whether or not the curve, the

function f(x), the right hand side of the
differential equation, when we're

drawing those phase lines if it crosses
the axis like this or like this.

Basically what that means is that it
depends on the slope.

Is the slope positive?
Is the slope negative?

There's an analog in 2D but it has to do
with the eigenvalues of a 2x2 matrix

instead of a simple slope.

Anyway, this is in most
differential equation textbooks.

Certainly those that take a modern
approach or a dynamical systems

approach which is more and more
textbooks these days.

If you want to know more it won't be hard
to find additional references.

So that was two dimensional
differential equations.

Then we looked at two
dimensional iterated functions.

These are like the logistic equation,
an iterated function.

Time moves in jumps, the population moves
in jumps. But now we just have two things

we're keeping track of: x and y.

This says the next value of x is this
function of the current value of y and x.

The next value of y in this case is
a function only of x.

This is an example of known as a Hénon
map; a commonly studied example.

The story for 2D iterated functions is
pretty similar to that of 1D.

There can fixed points, there can be
cyclic behavior and there can be aperiodic

behavior, there can be chaos.

What's new is that we can plot these x and
y points on a 2D grid like I've done here.

That's another way to see the sorts
patterns that the orbits are creating.

We'll spend a bunch of time on that in
the next unit.

So just a little bit more about two
dimensional iterated functions.

Mathematically they're very similar to 1D
iterated functions. Orbits can be periodic

and we can also have chaotic behavior.

Just a reminder that chaotic behavior is
bounded, aperiodic orbits that have

sensitive dependence
on initial conditions.

Lastly, we looked at three dimensional
differential equations.

The example I used was the Lorenz
equations.

Here are three differential equations.
The variables are x, y, and z.

This is a very simplified model of
atmospheric convection.

The physical origins of this model aren't
important for what we want to do here.

There are three parameters. They are Greek
letters: σ (sigma), ρ (rho), β (beta).

So this is a rule for how three things
vary in time instead of two.

If we use Euler's method, or the like, to
get a solution for this we'll get three

solution curves; x vs. time,
y vs. time, z vs. time.

We can plot those in time,
and in this case we see

periodic wiggling in all of these.

Then it can be interesting and fun
to plot x, y, and z against each other.

So this is like a phase plane but because
there is a z now it's phase space, it's

three dimensional. That makes things a
lot more interesting.

So here's what those periodic solutions
look like in phase space.

So it's oscillating up and down in the z
direction, sideways in the x, and sideways

in the y and it makes this curly, loopy
shape.

It looks like these lines cross but they
don't really because this is three

dimensional space so this line is above
that one. So just like this finger is

above that one. It looks like they cross
but they don't.

They aren't really because
it's a three dimensional space.

So curves in phase space cannot intersect,
just like they can't intersect on the

phase plane. But because space is three
dimensional curves can go over or under

each other. That means for three
dimensional differential equations the

solution curves can sort of wind around
each other and weave around each other

without actually crossing. So we'll see
much more complicated behavior in 3D

differential equations than in 2D.

In particular, three dimensional
differential equations can be chaotic.

We'll explore chaos in these 3D
differential equations and 2D iterated

functions in the next unit.

Those explorations will mostly be in phase
space so we'll be looking at what things

look like in phase space and we'll be in
for some pretty fun surprises.

So this brings us to the end of Unit 7.

In this unit we've looked at dynamical
systems in two and three dimensions.

In a certain sense the story is the same
as in one dimension.

A dynamical system, remember, is just a
deterministic rule that describes how a

quantity changes in time.

For two and three dimensional systems we
are just keeping track of two or three

quantities instead of one.

Mechanics of finding those solutions is
essentially the same no matter how many

variables we're trying to keep track of.
We just follow the rule. That's what

the dynamical system tells us to do.

However, visualizing solutions for higher
dimensional systems is a little more

interesting and fun because, well, two and
three dimensions are a little more

interesting and fun than one dimension.

In particular, for differential equations
in higher dimensions we see much more

complex and interesting behavior.

That'll be the topic of the next unit,
Unit 8. Which is on strange attractors.

We'll see these incredible structures that
combine order and disorder in some

really interesting and surprising ways.

We'll see you next week in Unit 8.