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affinity). They are called renormalization group equations, and, with respect to prediction, they replace any or all of the original specific predictive equations for the particular system under study (Cvitanovic', 1989). Whereas the U sequence and critical point behaviors are manifestations of qualitative universality, these scaling numbers are manifestations of quantitative universality. We discuss them here because their omnipresence in computationally realized differential equations as well as physical and chemical experiments along with their quantitative specificity (with values in all systems as “constant” as π) constitute a most persuasive argument for the substantiality of modern dynamical systems approaches to brain and other biological research.

The physical and physiological requirements for manifestations of these universal bifurcation scenarios can appear to be remarkably minimal. In physics, for example a full panoply can be observed in a “dripping faucet” (Shaw, 1984). Similarly, a small piece of extirpated and perfused myenteric or femoral artery will demonstrate these transitions in vasomotion spontaneously and almost independent of flow rate (Stergiopulos et al, 1998).

Feigenbaum discovered that in dynamical systems manifesting a series of period doubling bifurcations, the ratio of the parameter value at which the next period doubling bifurcation occurred relative to the last one ≈ 1/4.6692 ... and the ratio of the magnitude of the spawning point to the one spawned ≈ 2.5. (Feigenbaum, 1979). By “rescaling” distances along a parameter value (see below) using what is called a “universal renormalization operator” the geometric situation around each bifurcation point (though of different absolute “size) remains relatively the same. In intermittent systems, burst length varies as the inverse square root of the distance of the value of the parameter from that value that elicited the fixed point (Manneville and Pomeau, 1980). The universal characteristic of the third common parametric route to chaos, quasiperiodicity, is that the ratio of independent frequencies found most resistant to mode locking and breakdown into chaos is, ω_i / ω_{i+1} = 1.618... the number to which the ratio of adjacent Fibonocci numbers converge (1, 2, 3, 5, 8, 13,

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