Stable set for $f(x) = \frac{\pi}{2}\sin(x)$

Question

So, i've been thinking about this question to find out what is the stable set for $f(x) = \frac{\pi}{2}\sin(x)\,,\,0\leq x \leq \pi$. More specifically, my question is: is there a way that we could represent the iterations of $\sin{(x)}$, for example $\sin{(\sin{(x)})}$, $\sin{(\sin{\cdots\sin{(x)}})}$, in the unit circle? That is to say, if we know that we are drawing a vertical line for the representation of $\sin{(x)}$ in the unit circle, what will happen with the subsequent lines representing further iterations of this particular function?

Answer 1

Assuming your angles are in $(-\pi, \pi]$, there are two stable points: $\pm \pi/2$ (and one unstable point: $0$). This can be seen by plotting $y=x$ and $y=f(x)=\frac{\pi}{2}\sin x$ on the same graph.

You can visualise the convergence in a staircase-like pattern (example) There is some theory on such convergences based on the first and second derivatives of $f$.

In this case it's easy: all $x\in (0,\pi]$ have $f^n(x)\to\pi/2$.