 Solution

For $r_0\gg L$, $\Omega /{\omega }_0\to 2$. The potential approaches the isotropic harmonic-oscillator form $k{r}^2/2$, and the perturbed path is approximately a small-eccentricity ellipse centered on the fixed point. The radius has two maxima and two minima per revolution. For finite nonzero $L/r_0$, the ellipse is an approximation and slow precession can remain.