I plotted several vector fields for different values of $r$, specifically: $$r = 1$$ This gave me half-stable fixed points at $2k\pi, k \in \mathbb{Z}$. $$r > 1$$ No fixed points. $$-1<r < 1$$ Infinitely many fixed points which change from stable to unstable $$r = -1$$ Half-stable fixed points at $(2k+1)\pi, k\in\mathbb{Z}.$ $$r < -1$$ No fixed points.
When attempting to plot the bifurcation diagram I get $x^* = \arccos(r)$, do I plot this on a restricted domain or an infinite domain?
How do I determine which values of $r$ give bifurcations as there are infinitely many of them, essentially I have no idea how to label the fixed points on the bifurcation diagram.
This is a non-generic situation, these are usually not named. Add some perturbation like $$ x=r-\cos(x)+\varepsilon\sin(\sqrt2 x) $$ to get all those fold-bifurcations isolated, one bifurcation per bifurcation point. \begin{align} 0&=f_r(x)=r-\cos(x)+\varepsilon\sin(\sqrt2 x)\\ 0&=f_r'(x)=\sin(x)+ε\sqrt2\cos(\sqrt2 x) \end{align} From the second equation $x_n\approx n\pi$ and iterating to get the next perturbation terms $$ x_n\approx n\pi-ε(-1)^n\sqrt2\cos(\sqrt2 n\pi) $$ and thus from the first equation $$ r_n\approx\cos(x_n)-ε\sin(\sqrt2 x_n) =(-1)^n(1+ε^2\cos^2(\sqrt2 n\pi)) -ε\sin(\sqrt2 n\pi)+O(ε^3) $$ which in general will not be equal for different values of $n$.