So far in this unit we've looked at two dimensional differential equations
and two dimensional iterated functions.
In this part of the unit I'll introduce three dimensional differential equations.
So now there'll be three variables to keep track of x ,y and z.
And the phase plane will turn into a phase space.
I'll introduce 3D differential equations through a system known as
the Lorenz Equations.
So let's get started.
The Lorenz equations are three differential equations, a three
dimensional system of differential equations.
They were introduced in the early 1960's by Ed Lorenz, an MIT meteorologist.
He proposed these as a very, very simplified model of convection
in the atmosphere.
He started with more complex models and stripped things away until he got these
three that were designed much as the other models we've looked at, to give some sort
of caricature or sketch of certain aspects of convection and maybe weather.
We won't worry about their physical origin we'll interpret them as three variables
and won't worry about what they mean: x, y, and z.
These equations have three parameters and the parameters are Greek letters
sigma (σ), rho (ρ), and beta (β).
This is a deterministic dynamical system like all the ones we've studied before.
Now we have three variables x, y, and z.
And given an initial condition, an (x,y,z) point, the future path of this
point is determined by these equations.
These directions tell x, y and z what to do.
And these three equations are coupled because for instance x depends on y,
y depends on x and z and so on.
So the evolution of these three variables are all related.
So I want to show solutions and discuss solutions to the Lorenz equations for
three different choices of parameter.
And the first choice I'm gonna do
there is a σ=10, ρ=9, and β=8/3.
So if I choose these three parameter values, put them in there,
choose an initial condition, an x value, a y value and a z value,
and then I can solve these equations produce solutions curves, using Euler's
method or in this case I used a Runge-Kutta method in Python, but same
basic idea and we can do that and we'll get three solutions.
We'll get a solution for x as a function of t.
So this is time, this is x. I think everything here's gonna start around 10.
So I started at 10 and x wiggles approaching a fixed point.
We can look at y, overlap these a little bit. There's y; y behaves quite similarly.
It again starts at 10 and then a wiggle and down like this.
Here is z; z looks a little different. Z also starts at 10 but it goes up first
and then does some wiggles as well.
In the two dimensional case, when we had rabbits and foxes, we had a rabbit curve
and a fox curve and we plotted rabbits against the foxes to look at the phase
plane picture, so we got time out and just wanted to look at how R and F,
rabbits and foxes, were related.
Here we have x,y and z, so if we want to get rid of time we'll have a three
dimensional plot x, y and z.
So again, just to be clear we can think of this as describing an x position where
we are moving this way.
This could be describing the y position.
And then this would be the z position, an altitude, how high or low the thing is.
So we plot these together it's a three dimensional thing.
It's hard to visualize three dimensions on a two dimensional piece of paper
but let's give it a try.
So here's a plot of those three solutions against each other.
So this is x, this is y, and this is z.
So x and y are both oscillating and z is oscillating as well.
So this would look, I'll exaggerate this a little bit. The motion would be
something like this and then it settles down and reaches a fixed point,
somewhere here.
So the point is, it's spiraling around in x and y and in z
and it ends up at a fixed point.
So this is one type of behavior we can see in three dimensional differential
equations like this. We have an attracting fixed point, and the orbit spirals in x, y
and z, it oscillates in x, y, and z as it approaches the fixed point.
Let's look at solutions to this equation for another set of parameter values.
So now σ will stay at 10, β will stay at 8/3 but now I'll have ρ be 160.
Let's see what solutions to that look like.
First here is the curve for x. This is a dynamical system.
We know the parameter values, we know the starting values that determines
future values, Euler's Method or something like it will produce this curve.
Start at 10, initially I spike up, down.
By the time I get to 2, 3, or 4 it's settled into some sort of a regular cycle.
It's not a perfect sine wave at all, it's got these funny wiggles in it.
But whatever it is, it's repeating.
So from here to here, is the same as from here to here, and so on.
Let's see what happens to y. Here's the y solution.
It also does some initial large wiggles but then settles into something that
repeats; from here to here is the same as from here to here.
It's a little bit odd looking. It looks molars and that looks like some
tooth and that's an upside down something, who knows what these look like.
The point is, it's repeating. It's repetitive.
It's gone into some sort of cycle.
And lastly let's look at z. Here's the z part of the solution.
So again some initial large wiggles but then those sort of shake out and we
get into some sort of cyclic behavior.
So this is repeated here, and it's repeated there.
So now we'd wonder, "what does this look like in three dimensional space
if I plot x, y, and z together?"
If I think about this as describing the x position of an object, this the y position
this the z position. What sort of curve would this trace out in space?
In order to get a better picture of that, I'm actually going to look, instead of
going from 0 to 6, I'm going to go from 10 to 16.
That'll give us a little bit of a better view of what the longer term behavior is.
So we've solved from 0 to 10 but we're only plotting from 10 to 16.
So again here in the x direction we can see these wiggles repeating.
Let's see if I can describe what this motion would be in the x direction.
So we start, x is negative (I'm starting right there) it moves up, it's positive,
moves down again, up more, down like this. This motion would be something like this.
Between my fingers is the point that would be moving around in this phase space.
Alright, so here's y. And that y motion would look something again regular wiggles
up and down. Regular in the sense of repeating not in the sense of straight
up and down like a sine wave might do. And then this would be z. So let's see.
It goes up far, and then down, and then up not as far, and then down. So this would
look something; up, [motion sounds]. So that's what this motion describes.
The total movement would be these three things all happening at the same time.
There's going to be an up [motion sounds] along with this motion in the x direction
and this in the y direction. And if we plot that in 3D, although projected onto a
piece of paper, it turns out that we get the following shape.
So it looks like this.
So it's moving up and down in z, it's moving side to side in x, side to side
in y and if you combine all of those together you get this closed structure.
So it repeats through space. I sort of imagine a fly, moving through a room,
it goes around like this and then spirals and goes back there.
So again, this is a periodic solution. Something I want to stress is that just
as was the case in two dimensions, in three dimensions lines in
phase space can't cross each other.
It looks like these are crossing each other but actually these lines this way
are on top of the line that's going that way. So it looks like my fingers are
crossing but they aren't really because there is a separation between them.
Reminder again, why phase lines can't cross, why you can't have things crossing
on a phase diagram is because that would violate the condition of determinism.
So if we imagine this object moving around in three dimensional space, what these
equations tell us is that there is a unique set of directions at each point
in space. If the lines crossed then we would have a non-unique direction that
would be violating this condition of determinism.
Here we have some complicated periodic behavior but it's still periodic,
it returns back to where it came from and once it loops back on itself it's doomed
to repeat forever. That's the nature of determinism.
Another way of saying that is this dynamic together with knowledge of a point in
space at any time determines the value of the curve in the future and in the past,
as it turns out.
So this is another type of behavior we can see in three dimensions, it's cyclic
behavior but notice it's more complicated in a sense than the cyclic behavior we saw
for two dimensional systems. Because it can appear to loop back on itself but it
doesn't really because of this three dimensional property.
So we've seen a fixed point and we've seen a periodic cycle, let's look at one more
type of behavior. Again, σ will be 10, β is 8/3 but now ρ will be 28.
These are the famous parameter values for this system.
These are the ones that Lorenz focused his papers on.
These parameter values, I specify the initial condition and that determines
a curve. Euler's Method is a way of figuring out what that curve is.
If we do that, we would get this. I think I showed you this curve before when we
were talking about chaos, back in Unit 3, I think. Seems like a long time ago.
So the thing to notice here is there is some sort of regularity to it, in that
there's wiggles that have some common spacing in terms of time but there's not
a regularity to the pattern. Here we have three downs, five ups, two downs, two ups,
three downs and so on. It sort of looks, there's some pattern here but it's
aperiodic, it turns out that this does not repeat.
We see a similar thing in y. There's some correlation between these two.
If you look carefully you'll see the curves are not exactly identical but they
are similar. Again, there's some regular behavior in terms of time, how these
things are spaced out over time, but the trajectory itself is aperiodic, it doesn't
repeat. Lastly we can look at the z solution. Here it is.
This looks different from x and y. Again note that it's not repeating.
So wiggles up and down, but the height the amplitude of these wiggles keeps changing.
Here's a bunch of short ones. Here it gets larger.
Up and down, up and down here, and so on.
So we can then do the same thing we did for the other sets of solutions.
Plot them not against time, but x against y against z in three dimensional phase
space and see what it looks like.
That's something that will wait until Unit 8.
You'll see that soon, something to look forward to for Unit 8 or you can go look
it up or code your own Lorenz equations up and see what that shape looks like.