Computer Algebra Recipes

8.5. RECONSTRUCTING AN ATTRACTOR 349
The points lie on the logistic curve, so we have been successful in identifying and
reconstructing Humpty Dumpty. We can con¯rm from either of the previous
two recipes that the logistic map produces deterministic chaos for a =3:9.
PROBLEMS:
Problem 8-27: Reconstructing the quartic map
Taking a =1:97 and x0 =0:9, iterate the quartic map
xn+1 = axn (1 ¡ x 3 n)
to produce a time series with 200 points. Plot the time series with a line
style. Attempt to reconstruct any possible underlying deterministic attractor
by plotting xn+1 versus xn using a point style. Show that these points lie on
the curve aX(1 ¡ X 3 )witha =1:97. (This problem is a bit like a dog chasing
its tail!)
8.5.2 Random Is Random
We humans have purpose on the brain. We ¯nd it hard to look at
anything without wondering what it is \for," what the motive for it
is, or the purpose behind it. When the obsession with purpose becomes
pathological it is called paranoia{reading malevolent purpose
into what is actually random bad luck. But this is just an exaggerated
form of a nearly universal delusion. Show us almost any object
or process, and it is hard for us to resist the \Why" question|the
\What is it for?" question.
Richard Dawkins, British biologist, author, River Out of Eden, 1995
In this recipe, we shall use Maple's random-number generator to produce a
time series and attempt to reconstruct any underlying deterministic attractor.
The time series will have N =200points.
> restart: N:=200:
The command randomize( ) sets the random-number seed to a di®erent value
based on the computer system clock.
> randomize():
In the following do loop, N random decimal numbers lying between 0 and 1 are
generated.
> for n from 0 to N do
The command rand( ) produces a random positive twelve-digit number. A
fractional number between 0 and 1 results on dividing by 1012 .Thenumberis
then put into decimal form by applying the °oating-point evaluation command.
> x[n]:=evalf(rand()/10^12);
> end do: