Rotating torsion-stabilized inverted pendulum ($\ddot{\theta }={\Omega }^2\sin \theta \cos \theta -\mu \theta /(m{a}^2)+(g/a)\sin \theta$)

A rigid pendulum tilted from the upward vertical and driven at fixed azimuthal angular velocity $\Omega$ has effective potential $\mu {\theta }^2/2+mga\cos \theta -m{a}^2{\Omega }^2{\sin }^2\theta /2$. With $\mu =2mga$, its upright equilibrium loses linear stability when ${\Omega }^2$ passes $g/a$. At equality the leading restoring torque is cubic and the potential has a positive quartic minimum. This gives a stable quartic oscillator with zero infinitesimal harmonic frequency, rather than a freely drifting neutral state.