 Homoclinic basin boundary in a sinusoidal radial flow

For $\dot{r}=r(1-r)(r-2\sin \theta )$, $\dot{\theta }=r$, the circle $r=1$ has normal Floquet multiplier $e^{-2\pi }$ and is attracting. Inside it, $w=(1-r{)}^{-1}$ satisfies $dw/d\theta =(1-2\sin \theta )w-1$. The solution through $w(\pi )=1$ has $w>1$ on one interval $({\theta }_b,\pi )$ with $-\pi <{\theta }_b<0$. The corresponding radius $r=1-1/w$ is a homoclinic orbit to the origin. Near its returning endpoint $r\sim (\theta -\pi {)}^2$, so the approach takes infinite physical time. Comparison for the scalar angular equation shows that smaller radii inside this loop approach the origin, whereas all positive radii outside it approach the circle. The origin is therefore not Lyapunov stable despite attracting an open set.