= Solution
The downward and upward positions $\theta=0,\pi$ are equilibria. Linearization about $\theta=0$ gives
$$
\ddot\eta=(\omega^2-g/R)\eta,
$$
so the downward equilibrium is stable for $\omega^2<g/R$ and unstable for $\omega^2>g/R$; at equality it is marginal at linear order. At $\theta=\pi$, the linear coefficient is $\omega^2+g/R>0$, so the upward equilibrium is unstable.
When $\omega^2\geq g/R$, there are also the two equilibria
$$
\theta=\pm\cos^{-1}\left(\frac{g}{R\omega^2}\right)
\pmod{2\pi}.
$$
At either nonvertical equilibrium, differentiation of the right-hand side gives $-\omega^2\sin^2\theta<0$, so both are stable when distinct. This is a <pitchfork bifurcation of a rotating hoop bead> at $\omega^2=g/R$.
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