These are animations of the system \(\dot{r}=r(1-r^2), \dot{\theta }=1\) which exhibits a stable limit cycle of radius 1.

And animation of the system \(\dot{r}=-r(1-r^2), \dot{\theta }=1\) where the limit cycle now becomes unstable.

Stable limit cycle. \(\dot{r}=r(1-r^2), \dot{\theta }=1\) |
Unstable limit cycle. \(\dot{r}=-r(1-r^2), \dot{\theta }=1\) |