Abstract
In a previous paper [Murphy and Renie, 1984] we noted that in a system that exhibits homeostasis of zero order, an instantaneous perturbation, however small, invariably leads to a permanent, stable oscillation, the amplitude of which depends on the restoration constant (b) and the lag time (L) but not on the magnitude or sign of the original perturbation. We suggested that such a remarkable property, so easily attained, would be a valuable and plausible evolutionary device for generating those cyclical processes so important in normal physiology β such as respiration, heart contraction, the sleep cycle, seasonal effects, and menstruation. By a suitable choice of parameters, they could be readily tuned to the requisite length of the cycle and amplitude of variation. The most immediate argument against this suggestion is the implausible form of the wave generated in the zeroβorder process, which is angular and quite unlike anything in common experience of normal physiology.
Then in a subsequent study [Renie and Murphy, 1984] we found that for homeostatic processes of higher order, in general there is a critical value of the restoration constant at which regular oscillation is maintained. A preliminary study suggested that for processes of fractional order, no critical value that produces oscillation exists. However, the topic of processes of fractional order seems to be of sufficient complexity to warrant detailed treatment in a separate paper.