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nonharmonic frequency encoding parameter spaces fusing, resulting in new periods that are the sums of their adjacent ones: period 2 + period 3 = period 5, in what is called the “period adding route”. Technically precise classification of bifurcations involve much more careful definitions and well studied technical constraints involving such issues as the symmetries and dimensionality of the system of observables, how many control parameters (“codimensions”) are required to reasonably realize the bifurcation and the particular way the fixed points of the system become unstable, all of which are directly explorable when the equations are known or can be hypothetically inferred from the qualitative behavior of real data.

We note a few examples from the wide variety of bifurcating systems that can be found in the biomedical literature of interest for the biological sciences. With substrate input rate as the bifurcation parameter, the phosphofructokinase regulated glycolytic cycle in yeast extract was found to change among steady state, periodic and period doubling (subharmonic) regimes (Boiteux et al, 1975). Transitions between steady state, oscillatory and chaotic patterns have been reported in variety of physiological measures in man including respiratory rhythms and circulating blood cell concentrations over time (Mackey and Glass, 1977; Glass and Mackey, 1988 ) and models of dopamine cell dynamics (King et al, 1984). Flow rate parameter sensitive periodic, bursting and chaotic behavior has been found in a peroxidase-oxidase system (Olsen and Degn, 1977). A brain enzyme, substantia nigral dopaminergic tyrosine hydroxylase, manifested different saturation and fluctuation patterns, including bursting and periodicity, in experiments in which low (physiological) levels of tetrahydrobiopterin cofactor were the bifurcation parameters and adrenergic drugs were used as modulators (Mandell and Russo, 1981).

All four of the generic bifurcation routes to chaos, period doubling, changing multifrequency (quasiperiodicity), period adding and bursting (called “intermittency”) were observed in self-sustained oscillations induced in the neural membranes of space clamped, giant squid axons that were immersed in a 550mM NaCl, and electrically stimulated over changing amplitudes and frequencies (Aihara et al, 1986; Takahashi et al, 1990). With external stimulus current level as the control

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