= Basin estimate from a Lyapunov sublevel set
Suppose a compact connected component $C$ of $\{x:V(x)\leq c\}$ contains an equilibrium $x_*$ and $\dot V<0$ throughout $C\setminus\{x_*\}$. Then $C$ is an <invariant sublevel set>, and every trajectory in $C$ converges to $x_*$ by the <LaSalle invariance principle>. Thus the interior of $C$ lies in the <basin of attraction> of $x_*$. The largest such component supplies the strongest basin estimate available from that Lyapunov function alone.
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