Looking for some help understanding the following concept. I know when you are trying to determine the stability of fixed points for the system,
$$x'=\sin x$$
You proceed with the following steps,
$$x'=0$$ $$\sin x=0$$ $$x^{*}=k\pi$$
deriving $x'$ to determine stability would give,
$$\cos x$$
plugging in the fixed point,
$$\cos (k\pi)$$
Therefore, when $k\pi$ is even it is equal to $1$, so it would be unstable. On the other hand when $k\pi$ is odd it is equal to $-1$ so it would be stable.
However what if you were given $$x'=\cos x$$.
The derivative would be $-\sin x$ and the fixed points would still be $k\pi$ but when $k\pi$ is odd and when it is even, is it stable or unstable?