Computer Algebra Recipes

8.5. RECONSTRUCTING AN ATTRACTOR 347
Although the data appears to be con¯ned to the approximate range 0 to 10, it
is not clear at ¯rst glance whether it represents deterministic chaos or is simply
a \noisy" set of data. Can we con¯rm that it is the former and identify the
probable identity of the underlying map? Calling this the \Humpty Dumpty
map," can we then reconstruct his or her mathematical \appearance"? Or, at
the risk of sounding rather melodramatic, can we put Humpty Dumpty back
together again?
The number of operands command reveals that there are 200 entries in x.
> N:=nops(x);
N := 200
First, let us plot x versus n and see whether that reveals any pattern that we
might have missed. By default, the N points are joined by straight lines.
> plot([seq([n,x[n]],n=1..N)],labels=["n","x[n]"]);
x[n]
8
6
4
2
0 50 100 150 200
n
Figure 8.15: Time series x versus n.
Although there is a hint of some repetition, it certainly isn't conclusive that the
series comes from a map producing deterministic chaos rather than just being
noisy output. Following the procedure mentioned in the introduction, we will
plot xn+1=10 versus xn=10, the factor of 10 being introduced to reduce both
vertical and horizontal ranges to 0 to 1 in the graph. The points are represented
by size-12 black circles. The graph is assigned a name, gr1, soitcan¯rstbe
displayed and then used to reveal Humpty Dumpty's true identity.
> gr1:=pointplot([seq([x[n]/10,x[n+1]/10],n=1..N-1)],
symbol=circle,symbolsize=12,color=black,
labels=["x[n]","x[n+1]"]):
Entering gr1 with a command-line-ending semicolon,
> gr1; #humpty dumpty?