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made on their attractors when they were much more densely filled (Lathrop and Kostelich , 1989). Bowen’s “shadow lemma” in support of a thin film of points over the skelton of unstable fixed points of attractors is the fundamental reason that short sample length time series can often discriminate between control and experimental conditions in brain research studies.

Another recently implemented entropy, called “approximate entropy,” is exploiting the underlying unstable fixed point skeletal shadowing principle in expansive dynamical systems to find statistically significant differences between control and experimental results in reasonably short, physiologically realistic, sample lengths (Pincus, 1991; Pincus et al, 1991). This algorithm is somewhat derivative of those involved in the computation of the correlation dimension (see above). Instead of computing across a range (and taking the limits) of embedding dimensions, d, and sequential paired-vectorial distances, ε, it empirically tailors and fixes them to compute a “logarithmic likelihood” that points remains close through incremental change in the time series. The “approximate entropy” is not easily relatable to either hT and hM. One is tempted to predict that this geometrically oriented algorithm might be fooled into a postive entropy diagnosis if applied to strange, nonchaotic dynamical systems with fractal dimension but no λ̄(+)-related mixing. Since sequence position is conserved in this computation, two simultaneously studied (“multiparameter”) systems can be examined for their mutual coherence as the “cross approximate entropy.” Among the interesting findings from applications of this index to neuroendocrine studies are an increase in approximate entropy in LH and FSH secretory patterns with age in both sexes, perhaps quantitatively heralding menopause (Pincus and Minkin, 1998) and decreased cross approximate entropy, a decrease in regulatory coupling between ACTH and cortisol secretion patterns in patients with Cushing’s syndrome (Roelfsema et al, 1998).

Among the many of other empirically derived entropies, one is called “power spectral entropy,” which is equivalent to the normalized variance of the distribution of frequencies in a power spectral transformation of a time series (Farmer et al, 1980). This has been successfully applied to brain enzyme and receptor fluctuations

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