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