Let $n\in\mathbb N$, let $U\subset \mathbb R^n$ be open, let $f:U\to \mathbb R^n$ be locally Lipschitz, and let $x^*\in U$ satisfy $f(x^*)=0$. For $y\in\mathbb R^n$ and $s>0$, write

\begin{align*} B(y,s):=\{z\in\mathbb R^n: |z-y|<s\}. \end{align*}

Consider the autonomous differential equation

\begin{align*} \frac{dx}{dt}=f(x). \end{align*}

Assume that $x^*$ is exponentially stable in the following explicit sense: there exist constants $r>0$, $C>0$, and $\alpha>0$ such that for every $x_0\in U$ with $|x_0-x^*|<r$, there exists a unique forward solution $\gamma_{x_0}\in C^1([0,\infty);U)$ satisfying $\gamma_{x_0}'(t)=f(\gamma_{x_0}(t))$ for every $t\geq 0$ and $\gamma_{x_0}(0)=x_0$, and this solution obeys

\begin{align*} |\gamma_{x_0}(t)-x^*|\leq C e^{-\alpha t}|x_0-x^*| \end{align*}

for every $t\geq 0$. Then $x^*$ is asymptotically stable in the following explicit sense: the constant map $\gamma_{x^*}\in C^1([0,\infty);U)$ with $\gamma_{x^*}(t)=x^*$ for every $t\geq 0$ is an equilibrium solution; for every $\varepsilon>0$ there exists $\delta>0$ such that for every $x_0\in U$ with $|x_0-x^*|<\delta$, there exists a unique forward solution $\gamma_{x_0}\in C^1([0,\infty);U)$ satisfying $\gamma_{x_0}'(t)=f(\gamma_{x_0}(t))$ for every $t\geq 0$ and $\gamma_{x_0}(0)=x_0$, and this solution satisfies

\begin{align*} |\gamma_{x_0}(t)-x^*|<\varepsilon \end{align*}

for every $t\geq 0$; and there exists $\rho>0$ such that for every $x_0\in U$ with $|x_0-x^*|<\rho$, there exists a unique forward solution $\gamma_{x_0}\in C^1([0,\infty);U)$ satisfying $\gamma_{x_0}'(t)=f(\gamma_{x_0}(t))$ for every $t\geq 0$ and $\gamma_{x_0}(0)=x_0$, and this solution satisfies

\begin{align*} \lim_{t\to\infty}|\gamma_{x_0}(t)-x^*|=0. \end{align*}