Computer Algebra Recipes

8.5. RECONSTRUCTING AN ATTRACTOR 351
There is no discernible pattern in the ¯gure, unlike the situation in the previous
recipe. By plotting other pairs, or even triplets, of numbers, one can conclude
(not surprisingly) that there is no underlying chaotic attractor here.
PROBLEMS:
Problem 8-28: Rolling a die
Consulting Maple's Help, produce a random set of 500 numbers from the positive
integers one to six inclusive. This might simulate the rolling of an honest
die. Make a plot of the \time series" and xn+1 versus xn to show the randomness.
Problem 8-29: Literature search
Apply the techniques of attractor reconstruction to some time series data (e.g.,
the Dow Jones index) extracted from newspapers, magazines, or whatever, and
see whether a pattern emerges.
8.5.3 Butter°y Reconstruction
A people's literature is the great textbook for real knowledge of them.
Thewritingsofthedayshowthequalityofthepeopleasnohistorical
reconstruction can.
Edith Hamilton, American classical scholar (1867{1963)
Our ¯nal recipe will illustrate how Lorenz's butter°y attractor can be reconstructed
from time series data extracted from the governing system of three
coupled nonlinear ODEs. The Lorenz system (which was discussed in Section
2.2.1) is entered, along with the initial condition x(0) = 2, y(0) = 5, z(0) = 5.
> restart: with(plots):
> sys:=diff(x(t),t)=sigma*(y(t)-x(t)),diff(y(t),t)=-x(t)*z(t)
+r*x(t)-y(t),diff(z(t),t)=x(t)*y(t)-b*z(t);
sys := d
x (t) =¾ (y(t) ¡ x (t));
dt
d
y(t) =¡x(t) z(t)+r x (t) ¡ y(t);
dt
d
z(t) =x(t) y(t) ¡ b z(t)
dt
> ic:=(x(0)=2,y(0)=5,z(0)=5):
The parameter values are taken to be r =28,b =8=3, and ¾ =10.
> r:=28: b:=8/3: sigma:=10:
The total number of entries in the time series will be taken to be N = 2000.
The total time is T = 50, so the step size ¢ = T=N is 1=40.
> N:=2000: T:=50: Delta:=T/N;