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.