The dynamics of quantifiable homeostasis. V: Homeostasis of higher order

The properties are explored of a cybernetic process with lag L and force of restoration equal to the size of displacement one lag unit earlier, raised to any arbitrary positive power (w greater than 1), multiplied by the restoration constant b, the sign being opposite to that of the displacement at that time. No closed-form solution is available, but a converging Taylor series expansion is presented that gives a solution of arbitrary precision. Unlike the linear system previously discussed, the amplitude of the displacement (A) is germane, and it is shown that the criterion that the effect of a perturbation be damped out is whether or not LbAw-1 exceeds a constant, kw, that depends only on w. The counterpart of this pattern of higher-order homeostasis is closer to the biological behavior of many forms of homeostasis than the scaled invariance characteristic of the linear process. kw is shown to be bounded between pi/2 and 2, and increases monotonically with w. Exactly computed values and large-sample approximations are given for the critical values below which no overshoot at all occurs. It is shown that where it does not, the minimum expected cost decreases with increasing w and reaches a minimum at infinite w. However, this evolutionary gain is vitiated where large displacements are involved, and may need to be redressed by a change in the restoration constant. Thus, the dynamic balance becomes much more subtle than with lower-order processes. Several brief clinical illustrations of these ideas are presented.