Consider the following system in polar coordinates $$\frac{dr}{dt}=(r^2-1)(2 r \cos(\phi)-1), \\ \frac{d \phi}{dt}=1.$$ The question is if the limit cycle $r=1$ is stable, and what is the region of attraction.
Simulations show that this limit cycle is attractive for $0<r<1$, and for $1<r<r+\epsilon$ with sufficiently small $\epsilon>0$. However, I cannot establish a strict proof and analysis.