Lyapunov 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 Lyapunov stable if for each $\varepsilon > 0$ there is a $\delta > 0$ with this property. For each $x_0 \in U$ with $\|x_0 - x^*\| < \delta$, the solution $\varphi_t(x_0)$ exists for all $t \ge 0$, and $\|\varphi_t(x_0) - x^*\| < \varepsilon$ for all $t \ge 0$. The point $x^*$ is unstable if it is not Lyapunov stable.