In this subunit, we'll go back and look again at the Lorenz equations.
At the risk of giving away the punchline, we'll see a strange attractor there too.
It's similar in some ways to the strange attractor of the Hénon map but there's
some differences too. Mainly because this is a three dimensional differential
equation and so the solution of the differential equation moves around
continuously in a three dimensional phase space. In contrast for the Hénon map, the
points jumped from position to position on a plane. But they're similar in that
it's an attractor on which the motion is chaotic.
So I'll start by reminding you about the Lorenz equations and showing solutions
for a few different parameter values. Then we'll use a couple different computer
programs to visualize the three dimensional structure of the Lorenz
attractor and see how it's both attracting and chaotic.
So here are the Lorenz equations. They're a system of three differential equations.
The variables are x, y, and z. There are three parameters σ, ρ, and β.
Once we fix those parameters we have a dynamical system and that's just a rule
that tells us how x, y, and z change over time.
Since this is a differential equation it is one of those indirect rules that
specifies x, y, and z by telling us, not x, y, or z directly, but their rates of
change.
So the solution to the Lorenz equations, we need to choose initial conditions,
initial x, y, and z. Of course fix parameters, σ, ρ, and β. And then use
Euler's method or something like it to plot some solution curves.
So let me just review what we did before. Here are solution curves x, y, and z for
these parameter values. So σ=10, ρ=9, and β=8/3. Each of the solutions wiggles up
and down and is approaching a fixed point; x is a little more than 5, 4 excuse me,
y is a little more than 4, z is right around 8. So if we plot these in three
dimensional space, so x against y against z, we're not going to plot time but just
how x depends on y and z and so on. We get the following picture.
Here it is. So x is wiggling as it approaches this fixed point. Y goes back
and forth as it approaches a fixed point. And z moves up and down as we appraoch
that fixed point. So we're spiraling in to a fixed point. So we saw that behavior,
we also saw some interesting periodic behavior, at least I think it's
interesting. So that was this parameter values. The only change is that ρ=160.
And we solve this dynamical system we get, there's our x curve, there's our y solution
and there's the z solution. So three variables, x, y, z, we get three solutions
x, y, z. And as before we think of this as specifying a point in three dimensional
space.