 Gravity-restored centrifugal governor bifurcation

For total ball mass $m$, arm length $l$ and imposed constant shaft speed $\Omega$, the reduced effective potential is $V_{\mathrm{eff}}/(m{l}^2)=-{\Omega }^2{\sin }^2\theta /2-{\omega }_0^2\cos \theta$. Its equilibrium equation is $\sin \theta ({\omega }_0^2-{\Omega }^2\cos \theta )=0$. The downward state is stable for $|\Omega |<{\omega }_0$ and loses linear stability above it; nonzero stable openings then satisfy the displayed relation. Extending the angle to signed values gives a supercritical pitchfork bifurcation. Squared angular frequencies of small oscillations are ${\omega }_0^2-{\Omega }^2$ at the downward state and ${\Omega }^2-{\omega }_0^4/{\Omega }^2$ at the open states. At threshold the quadratic term vanishes but the quartic potential is positive, so zero linear frequency does not imply finite-amplitude free motion.