Computer Algebra Recipes

346 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS
so that xn+1 depends only on the previous value xn. If this is the case, then a
functional form f will exist such that xn+1 = f(xn).
Assuming that this is the case, one plots pairs of numbers (xn; xn+1) from
the time series to form the two-dimensional space xn+1 versus xn. Ifthepoints
appear to lie on a de¯nite geometrical line, this implies that there is an underlying
attractor and an associated functional form f. If the geometrical shape
has more structure to it, this could imply that there is an underlying \twodimensional"
map, namely,
xn+1 = f(xn)+g(xn¡1); or xn+1 = f(xn)+yn; yn+1 = g(xn):
If the dimensionality of the underlying map is higher than two, one must increase
the dimensionality of the space accordingly. To \see" a three-dimensional
map, for example, one must work in a three-dimensional space. The above procedure
is then generalized by plotting triplets of numbers. For example, one
might use (x0; x1; x2), (x1; x2; x3), etc.
8.5.1 Putting Humpty Dumpty Together Again
Humpty Dumpty sat on a wall, Humpty Dumpty had a great fall,
All the king's horses, And all the king's men,
Couldn't put Humpty Dumpty together again.
Lewis Carroll, Alice's Adventures in Wonderland, 1865
Suppose that we have been given the following lengthy data list x, whereeach
entry x(n) corresponds to a time t = nts, withn =1; 2; :::;N.
> restart: with(plots):
> x:=[6.24, 9.15, 3.03, 8.24, 5.66, 9.58, 1.56, 5.14, 9.74, .980,
3.45, 8.81, 4.09, 9.43, 2.10, 6.47, 8.91, 3.79, 9.18, 2.93, 8.07,
6.07, 9.30, 2.53, 7.37, 7.56, 7.18, 7.89, 6.50, 8.88, 3.89, 9.27,
2.65, 7.60, 7.12, 8.00, 6.24, 9.15, 3.03, 8.23, 8.23, 5.67, 9.57,
1.60, 5.23, 9.73, 1.03, 3.59, 8.98, 3.58, 8.97, 3.61, 9.00, 3.51,
3.86, 9.24, 2.74, 7.76, 6.77, 8.53, 4.89, 9.74, .969, 8.89, 3.41,
8.77, 4.21, 9.51, 1.82, 5.81, 9.50, 1.87, 5.93, 9.41, 2.16, 6.61,
8.74, 4.30, 9.56, 1.64, 5.33, 9.71, 1.11, 3.85, 9.23, 2.76, 7.80,
6.70, 8.62, 4.65, 9.70, 1.13, 3.91, 9.28, 2.59, 7.48, 7.34, 7.61,
7.10, 8.04, 6.16, 9.23, 2.78, 7.83, 6.62, 8.72, 4.34, 9.58, 1.56,
5.13, 9.74, .977, 3.44, 8.80, 4.13, 9.45, 2.02, 6.29, 9.10, 3.18,
8.46, 5.08, 9.75, .960, 3.39, 8.73, 4.31, 9.56, 1.62, 5.30, 9.72,
1.08, 3.76, 9.15, 3.04, 8.25, 5.63, 9.60, 1.51, 5.00, 9.75, .951,
3.36, 8.70, 4.43, 9.62, 1.42, 4.75, 9.73, 1.04, 3.63, 9.02, 3.45,
8.81, 4.09, 9.43, 2.10, 6.47, 8.90, 3.81, 9.20, 2.88, 8.00, 6.23,
9.16, 3.01, 8.21, 5.73, 9.54, 1.70, 5.49, 9.66, 1.30, 4.41, 9.61,
1.45, 4.84, 9.74, .989, 3.48, 8.84, 3.98, 9.35, 2.38, 7.07, 8.07,
6.07, 9.30, 2.52, 7.35, 7.59, 7.14, 7.97, 6.30, 9.09, 3.23, 8.53,
4.88, 9.74, .972, 3.42]: