Transform the system
$$x' = y - x(x^2+y^2-1)$$
$$y' = -x - y(x^2+y^2-1)$$
to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all trajectories (except the singular point) tend to it as $t \to \infty$. Show therefore that the origin is an unstable focus.
What I tried:
The system becomes
$$r' = r(1-r^2)$$
$$\theta' = -1$$
For $0<r<1$, $r'<0 \to (0,0)$ is an unstable focus.
For $r>1$, $r'>0 \to (0,0)$ is a stable focus.
So I'm guessing we have to show that $r>1$ is problematic? How is it problematic?