This short presentation considers the logistic function

$\sigma \left(x\right)=\frac{1}{1+e^{-x}}$

and some extensions. This function can be defined by the differential equation

${\sigma }^{\prime }=\sigma (1-\sigma )$

which has the general solution

$\begin{array}{c}\frac{{\sigma }^{\prime }}{\sigma (1-\sigma )}=\frac{{\sigma }^{\prime }}{\sigma }+\frac{{\sigma }^{\prime }}{1-\sigma }=1\\ ln\sigma -ln(1-\sigma )=x-lnc\\ \frac{\sigma }{1-\sigma }=\frac{e^x}{c}\to \sigma =\frac{1}{1+c{e}^{-x}}\end{array}$

where the constant of integration allows additional choices for the function. This is how they look as the constant of integration is varied:

One can also view pairs of the functions together for positive and negative values of the constant of integration,

where only the upper portion of the function with the negative constant is shown. The reason for doing this is that by replacing the regular plots with polar plots,

one has a simple model of approach to a limit cycle. The circular limit is approached for both cases as $x\to \infty$ , spiraling out from the origin at $x\to -\infty$ for a positive constant of integration and coming in from infinite radius at $x=-ln(-\frac{1}{c})$ for a negative constant of integration.

The line approached at infinite radius for a negative constant of integration displays some interesting behavior. While its slope is in general given by $tan[-ln(-\frac{1}{c}\left)\right]$ , for special values of the constant it becomes either a horizontal or vertical line with a noticeable offset from the origin of unity. The former occurs when $-ln(-\frac{1}{c})$ is equal to integral multiples of $\pi$ and the latter when equal to integral multiples of   $\frac{\pi }{2}$ . This behavior can be explored by altering the code for the previous interactive graphic to allow a closer approach to the singularity generating the behavior.

Note that the limit cycle here is exactly circular because both functions approach unity at infinity. The graphic does not preserve aspect in order to accommodate the given range of input values.

One way to generalize the logistic function is by altering the differential equation to read

${\sigma }^{\prime }=\sigma (1-{\sigma }^n)$

where the added exponent is an arbitrary number. The general solution becomes

$\begin{array}{c}\frac{{\sigma }^{\prime }}{\sigma (1-{\sigma }^n)}=\frac{{\sigma }^{\prime }}{\sigma }+\frac{{\sigma }^{\prime }{\sigma }^{n-1}}{1-{\sigma }^n}=1\\ ln\sigma -\frac{1}{n}ln(1-{\sigma }^n)=x-\frac{1}{n}lnc\\ \frac{{\sigma }^n}{1-{\sigma }^n}=\frac{e^{nx}}{c}\to \sigma =\frac{1}{(1+c{e}^{-nx}{)}^{1/n}}\end{array}$

where the constant of integration has be chosen for a simpler result. The fractional root here implies that the function becomes complex for $x<-\frac{1}{n}ln(-\frac{1}{c})$ , and that must be taken into account when reprising interactive graphics with the generalized function.

Viewing pairs of the functions together for positive and negative values of the constant of integration,

where only the real portion of the function with the negative constant is shown, one can see that both functions approach the limiting value of unity more quickly as n is increased. Again replacing the regular plots with polar plots

one sees that the limit cycle occurs more readily as n is again increased.

Other generalizations include additional constants that can be varied. All of the graphics on the referenced page could be repeated here as interactive graphics.

Uploaded 2020.06.16 analyticphysics.com