Nonlinear Systems & Linearization · Challenge problem

A bead slides without friction on a vertical hoop spinning fast about its vertical diameter. Its angle $x$ from the bottom obeys $x' = y,\; y' = \sin x\,(2\cos x - 1)$. The bottom $x = 0$ is unstable, and the bead can rest stably at $x = \tfrac{\pi}{3}$. Find the period of small oscillations about $x = \tfrac{\pi}{3}$.