$$\dot x = -2x-y^2$$ $$\dot y = -y-x^2$$
$(0,0)$ is an obvious attractive fixed point, and I'll only look at this one.
I need to get the maximal radius $r > 0$ for a ball centered on the origin so that ball is still included in the basin of attraction.
I used $x^2+y^2$ as my Lyapunov function and determined that its domain of validity, defined by $\dot L < 0$, is given by $ x > -1$ and $y > -2$. But since it is an arbitrary choice I'm not sure I can conclude something from that.