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Rössler’s generic chaotic system (see above) moving recurrently in a three dimensional box can be orthogonally decomposed into three directional motions in a moving frame, each with a signatory sign of λ. The unstable direction of expansive stretching is characterized by some number > 0, λ(+), the stable direction of contractive folding, some number, < 0, λ(−), and the neutrally stable direction of recurrence, λ(0). For The “Lyapounov spectrum” of the Rössler attractor is [λ(+),λ(−),λ(0)] (Shaw, 1981). An n-dimensional dynamical systems has n one-dimensional Lyapounov exponents, and it is sometimes the case in relatively noise free, finite semi-stationary data lengths of the neurosciences, that a λ > 0 can be shown to exist for a second one, in a dynamical situation called “hyperchaos” by (Rossler, 1979). For example, two and sometimes three λ(+) have been reported in the flows on the EEG attractor of normal alert subjects (Gallez and Babloyantz, 1991). The presence of measurement noise, the finiteness of neurophysiological sample lengths as well as the relatively small expansive actions in some directions in the chaotic attractors of brain dynamics lead to the finding that most often, only one “leading Lyapounov exponent,” λ(+), is reliably computable (Sano and Sawada, 1885; Wolf et al, 1985; Eckmann et al, 1986).

A counter-intuitive fact about the stability of a dynamical system when a decrease in the value of λ(+) is observed such that λ(+) → λ(0) , is that this more neutral stability augers a global bifurcation (Guckenheimer and Holmes, 1983). A small perturbation does not change the global dynamics of an already expanding and contracting (called “hyperbolic”) dynamical system, it will maintain the style of its motions. However, when λ(+) → λ(0) , a velocity changing perturbation evokes a bifurcation to a new dynamic in what is called “loss of hyperbolic stability.” The best examples come from the observations of this kind of change in the EEG predicting the onset of epileptic seizures in patients with focal or temporal lobe epilepsy (Iasemidis et al, 1988,1990; Iasemidis and Sackellares, 1996 ).

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