Exact anisotropic radial limit cycle ($r_p(\theta )=[2-(\cos 2\theta +2\sin 2\theta )/5{]}^{-1/2}$)

For $\dot{r}=r[1-r^2(2-\cos 2\theta )]$, $\dot{\theta }=2$, the substitution $w=r^{-2}$ gives $dw/d\theta +w=2-\cos 2\theta$. Its positive periodic solution yields the displayed orbit, with full planar period $\pi$. Every other solution at the same phase has $w-w_p=C{e}^{-2t}$, proving uniqueness and attraction. The limiting circles $r=1/\sqrt{3}$ and $r=1$ give a trapping annulus, with tangencies at their extremal angles.