Context: Consider a hoop of radius $R$ fixed in a vertical plane. Given a bead of mass $m$ placed on the hoop, assuming that it remains fixed on the hoop and is allowed to slide without friction, it can be shown that the bead satisfies the idealized pendulum equation $\ddot\vartheta=(g/R)\sin\vartheta$, where $\vartheta$ is the angle from the top of the hoop to the position of the bead as measured from the center of the hoop. It can also be shown that the acceleration $\ddot{\mathbf{r}}$ of the bead is given as a function of $\vartheta$: $$\ddot{\mathbf{r}}(\vartheta)=\begin{pmatrix} g\sin\vartheta\cos\vartheta-R\dot\vartheta^2\sin\vartheta \\ -g\sin^2\vartheta-R\dot\vartheta^2\cos\vartheta \end{pmatrix}$$ Suppose we want to locate the points where the acceleration of the bead is exactly vertical. This occurs when the $x$-coordinate of $\ddot{\mathbf{r}}$ vanishes, or equivalently, where $g\sin\vartheta\cos\vartheta=R\dot\vartheta^2\sin\vartheta$. Solutions can be shown to occur when $\vartheta=0$ and $\vartheta=\pi$, or when the bead is at the top and bottom of the hoop, as is physically intuitive. However, we also have a solution when $\dot\vartheta=0$, or equivalently, when the angular velocity of the bead vanishes.
Question: How do we determine where $\dot\vartheta=0$?
The solution to the differential equation $\ddot\vartheta=(g/R)\sin\vartheta$ is given by the Jacobi amplitude function, and hence it suffices to find the critical points of this function. However, this seems to be a highly nontrivial task - not only is the Jacobi amplitude function difficult to work with in general, the solution appears to be very dependent on initial conditions. For high enough initial velocities, no critical points exist (the bead will continue rotating about the hoop indefinitely).
Any methods to solve for (or approximate) the critical points?