Basin estimate from a Lyapunov sublevel set

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

For $\dot{z}=Az+\Vert z{\Vert }^{2}z$, suppose a positive-definite matrix $P$ solves $A^{T}P+PA=-2I$. The quadratic function $V=z^{T}Pz$ then obeys $\dot{V}=2\Vert z{\Vert }^2(V-1)$. Its unit ellipsoid separates decay toward the origin from escape. In a planar system with nonzero angular speed it is an invariant periodic orbit.