Asymptotic stability

Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the flow of $x' = f(x)$, and let $x^* \in U$ be an equilibrium point. The point $x^*$ is asymptotically stable if it is Lyapunov stable and there is an $\eta > 0$ with this property. For each $x_0 \in U$ with $\|x_0 - x^*\| < \eta$, $\varphi_t(x_0) \to x^*$ as $t \to \infty$. If $U = \mathbb{R}^n$ and $\varphi_t(x_0) \to x^*$ for each $x_0 \in \mathbb{R}^n$, and $x^*$ is Lyapunov stable, then $x^*$ is globally asymptotically stable.