Sigmoids and the Seventh-year Trifurcation, a Metaphor

Abstract

Because the sigmoid is such an important function, it would be useful to put it in a more universal form, independent of parameters such as a, b and k, which are specific to the particular system under consideration, such as interest rate, amounts of reagents, half-lives, etc. We can scale the variables in equation (17-5) as follows. Instead of y we shall define a variable z which equals +1 when y = b, and which equals −1 when y = a; its aymptotic values would be z = − 1 and z = + 1. We write z = Ky + L and will determine K and L such that z has the desired asymptotes:

$$\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} { - 1 = Ka + L} \hfill & {and} \hfill & { + 1 = Kb + L,} \hfill & {hence} \hfill & {K = 2/(b - a),} \hfill \\ \end{array} } \hfill \\ {\begin{array}{*{20}{c}} {L = (a + b)/(a - b),} & {and} \\ \end{array} } \hfill \\ {z = (2y - a - b)/(b - a),y = \frac{1}{2}[(b - a)z + a + b].} \hfill \\ \end{array}$$