2.27 Lorenz Attractors
(type=lorenz/lorenz3d)
The "Lorenz Attractor" is a "simple" set of three deterministic
equations developed by Edward Lorenz while studying the non-
repeatability of weather patterns. The weather forecaster's basic
problem is that even very tiny changes in initial patterns ("the beating
of a butterfly's wings" - the official term is "sensitive dependence on
initial conditions") eventually reduces the best weather forecast to
rubble.
The lorenz attractor is the plot of the orbit of a dynamic system
consisting of three first order non-linear differential equations. The
solution to the differential equation is vector-valued function of one
variable. If you think of the variable as time, the solution traces an
orbit. The orbit is made up of two spirals at an angle to each other in
three dimensions. We change the orbit color as time goes on to add a
little dazzle to the image. The equations are:
dx/dt = -a*x + a*y
dy/dt = b*x - y -z*x
dz/dt = -c*z + x*y
We solve these differential equations approximately using a method known
as the first order taylor series. Calculus teachers everywhere will
kill us for saying this, but you treat the notation for the derivative
dx/dt as though it really is a fraction, with "dx" the small change in x
that happens when the time changes "dt". So multiply through the above
equations by dt, and you will have the change in the orbit for a small
time step. We add these changes to the old vector to get the new vector
after one step. This gives us:
xnew = x + (-a*x*dt) + (a*y*dt)
ynew = y + (b*x*dt) - (y*dt) - (z*x*dt)
znew = z + (-c*z*dt) + (x*y*dt)
(default values: dt = .02, a = 5, b = 15, c = 1)
We connect the successive points with a line, project the resulting 3D
orbit onto the screen, and voila! The Lorenz Attractor!
We have added two versions of the Lorenz Attractor. "Type=lorenz" is
the Lorenz attractor as seen in everyday 2D. "Type=lorenz3d" is the
same set of equations with the added twist that the results are run
through our perspective 3D routines, so that you get to view it from
different angles (you can modify your perspective "on the fly" by using
the <I> command.) If you set the "stereo" option to "2", and have
red/blue funny glasses on, you will see the attractor orbit with depth
perception.
Hint: the default perspective values (x = 60, y = 30, z = 0) aren't the
best ones to use for fun Lorenz Attractor viewing. Experiment a bit -
start with rotation values of 0/0/0 and then change to 20/0/0 and 40/0/0
to see the attractor from different angles.- and while you're at it, use
a non-zero perspective point Try 100 and see what happens when you get
*inside* the Lorenz orbits. Here comes one - Duck! While you are at
it, turn on the sound with the "X". This way you'll at least hear it
coming!
Different Lorenz attractors can be created using different parameters.
Four parameters are used. The first is the time-step (dt). The default
value is .02. A smaller value makes the plotting go slower; a larger
value is faster but rougher. A line is drawn to connect successive orbit
values. The 2nd, third, and fourth parameters are coefficients used in
the differential equation (a, b, and c). The default values are 5, 15,
and 1. Try changing these a little at a time to see the result.