Will it converge?

Let a series be defined, for some real value of $a_0$ and $\alpha(>0)$ as:

$\displaystyle \large a_n=a_{n-1}+\alpha \sin(a_{n-1})$

$X=\lim _{ n\rightarrow \infty }{ { a }_{ n } }$

Let $S$ be the sum of all the distinct values of $X$ as $a_0$ varies from $[1,100]$ and $\alpha$ varies over all positive reals.

Find $\left\lfloor 10^4S \right\rfloor$

Details and Assumptions:

1. $X$ must be finite.

2. $\alpha$ is a variable independent of $a_0$ and can take any positive value.

This is part of my set Powers of the ordinary .