= Quadratic-logarithmic Lyapunov function for a feedback system
{title2=$L=x^2-1-\log(x^2)+(y-1)^2$}
For $\dot x=x(1-y)$ and $\dot y=x^2-y$, on either half-plane $x\ne0$ 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 $x=0$.
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