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The number and variety of algorithmic strategies for computing Lyapounov exponents that are applicable to real data divide naturally into those that compute directly the average rate of separation of neighboring points from the "fiduciary" orbit, as observed on the reconstructed attractor, from which only the largest λ can be obtained (Wolf et al, 1985), and a variety of techniques based on assumed model maps of the unknown flow along which the sequential products of the local derivatives are computed. The logarithms of the straight line slopes of the sequence of directionally decomposed local tangent vectors multiplied, yield as many Lyapounov exponents as directions (Sano and Sawaka, 1985; Eckmann et al, 1986; Geist et al, 1990). The techniques of regularization by which these model processes approximate the unknown flow include those with least squares, linear fit assumptions (Eckmann et al, 1986; Sato et al, 1987; Buzug et al, 1990), more detailed fits involving polynomial expressions in higher powers (Briggs, 1990; Brown et al, 1991; Bryant et al, 1991) and techniques such as "singular value decomposition" which decomposes the flow into orthogonal components before computing the logarithmic rate of divergence of nearby points on each of them (Stoop and Parisi, 1991). A clever check on the Lyapounov number obtained is to study the flow backwards so that, for example, some rate of separation of points in the forward direction would approximate the rate of convergence in the time reversed data (Parlitz, 1992).

Among the sources of spurious Lyapounov exponents are sample lengths that are too short and/or too measurement-noisy to compute a statistically stable average, embedding dimensions that are too high or low and attractors (many of physiological relevance) that have geometric features such as sharp corners or tight folds as in the Rössler (where points gather) or delicate boundary points such as those on the body on the Lorenz butterfly (see above) where very small distances determine whether the orbit makes big jumps to the right or left wing leading to uncharacteristically large separations. This "nonuniformity" in the rates of expansion and contraction in the dynamics over the attractor, a source of error in computations of statistical indices of the average behavior, becomes a useful tool in characterizing individual differences in sets of neurobiological data ranging from

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