points) or more or less regular cycles. We analyze our “fixed point” data using

points) or more or less regular cycles. We analyze our “fixed point” data using quantities such as the mean and variance of distributional statistics and the cycle data using the amplitude, frequency, cycle length and phase of trigonometric functions. In central tendency-oriented research, rare, very high amplitude events have usually been considered aberrations and tossed, and imperfect periodic behavior is treated by “cosiner analysis” as regular cycles contaminated by measurement or system noise. Whereas technically, chaotic dynamics must live in dimension greater than two (for orbits to be more than a fixed point or limit cycle, able to snake around without necessarily intersecting ), the Lorenz attractor has dimension just a little over two, our difficulties with establishing the “true” physiological dimension of real biological observables (see below) makes such a consideration more theoretical than practical. The orbits of a forced-dissipative dynamical system in a parameter regime engendering chaos, converge onto an attractor which is neither a fixed point nor a limit cycle, thus the origin of the name “strange attractor” (Ruelle and Takens, 1971). It was James Yorke that first named these dynamics “chaos” (Li and Yorke, 1975). The necessarily statistical properties of the chaotic orbits on strange attractors follow from the generic characteristics of their motions (see Shaw, 1981 for a still conceptually current, non-mathematical treatment). These kinds of statistics are studied in a research context called the “ergodic theory of dynamical systems” (Ruelle, 1979; Eckmann and Ruelle, 1985). Ergodic is a word used to characterize a system with (or without) a particular condition placed on its statistical measures: the existence of an invariant measure which is undecomposabile into two invariant measures and, equivalently (though not obviously) one in which the time average equals its average in the geometric space into which it is embedded. One may arrive at the same ergodic measure from studying a single very long orbit or from summing across many individual but shorter orbits. This ergodic equivalence is made possible due to the definitional existence of at least one invariant statistical measure and the dynamics of the system which ideally include a uniformly, sequence disordering process called “mixing” (see below). 201 HOUSE_OVERSIGHT_013701