Quadratic-logarithmic Lyapunov function for a feedback system ($L=x^2-1-\log (x^2)+(y-1{)}^2$)

For $\dot{x}=x(1-y)$ and $\dot{y}=x^2-y$, on either half-plane $x\ne 0$ the Lyapunov function $L$ is nonnegative, has compact sublevel sets contained in that half-plane, and satisfies $\dot{L}=-2(y-1{)}^2$. The only invariant points of $\{y=1\}$ are $(1,1)$ and $(-1,1)$. The invariance argument therefore makes each half-plane converge to its corresponding stable focus. The stable set of the saddle at $(0,0)$ is exactly the line $x=0$.