Rapidly rotating saddle confinement (${\omega }_{\mathrm{slow}}\simeq {\omega }^2/(2|\Omega |)$)

In rapid rotation, both rotating-frame modes are nearly circular and counter-rotate relative to the axes at frequency near $|\Omega |$. Transforming back to the laboratory leaves a slow nearly harmonic oscillation and small rapid micromotion. For positive rotation the slow ellipse also has higher-order prograde precession at rate ${\omega }^4/(8{\Omega }^3)$. These follow by expanding $\sqrt{{\Omega }^2\pm {\omega }^2}$ and combining the two laboratory circular components.^[ref]