# Asymptotic stability without Lyapunov

Consider the following system of equations

$$\begin{cases} x'=-x^3y^2 \\ y'=-2x^2y^3 \\ \end{cases}$$ Using the Lyapunov function $V(x,y)=x^2+y^2$ we get $$\dot{V}(x,y)=-2x^4y^2-4x^2y^4\leq 0$$ So, $(0,0)$ is a stable equilibrium of the system. But what about asymptotic stability, can we say something about it? It is easy to see that every punctured neighborhood of $(0,0)$ contains points of the form $(\epsilon,0)$ and $\dot{V}$ vanishes on them. So, we can't use $V$ to get asymptotic stability.

The points $(\epsilon,0)$ are themselves equilibria, so clearly $(0,0)$ is not asymptotically stable.