Jacobi Elliptic sn – the case of a stuttered sinus

In my last post I introduced Jacobi elliptic sinus, the function sn(u,m), with u real, and 0\leq m\leq 1. One has to be careful with notation here. Let us have a look at the definition as it is given on Wolfram’s pages:

It is almost the same, but not the same as one from “Handbook of Mathematical Functions“, Ed. Milton Abramowitz and Irene A. Stegun:

Here Abramowitz and Stegun write simply sn u, but later on they use the notation sn(u|m), where m=k^2 of Wolfram. Wolfram’s Mathematica is using JacobiSN(u,m). Maple is using JacobiSN(u,k).
Matlab warns the user:

We will use sn(u,m).
The parameter k=\sqrt{m} is sometimes called the “modulus”. The shape of the function depends on the value of m. When m=0 we have just ordinary sinus:

At the other extreme, for m=1, sn is nothing else but the hyperbolic tangent:

In between, for 0<m<1, it is what I would dare to call a “stuttering sinus“. This stuttering is not seen at all for m=0.1 It looks just as if the period became a little longer:

and it is hard to notice for m=0.9

But for m=0.9999999 we get

It almost looks like the hyperbolic tangent, but when we zoom out we can see the stuttering:

Clear case of stuttering

The graphs looks somewhat like that of a rectangular signal. There are “flips” and then there are longer and longer periods when the function is almost constant – between the flips. This is the main characteristic of Dzhanibekov effect: there are almost pure rotation periods, and sudden flips when the axis suddenly reverses the direction. But for all values of m the function is periodic (for m=1 we have an exception – we have infinite period).

For m=0 the period is 2\pi, then slowly grows, but for m very close to 1 it becomes very sensitive to the value of m. For this reason every repetition of the Dzhanibekov effect as seen in the movies taken in space would probably give a different period.