Let $D \subset \mathbb R^n$ be an open neighbourhood of $0$, and let $f\in C^1(D;\mathbb R^n)$ satisfy $f(0)=0$. Suppose that $0$ is a locally asymptotically stable equilibrium of $\dot{x}=f(x)$: it is Lyapunov stable, and there exists $r>0$ such that $B(0,r)\subset D$, every solution with $|x_0|<r$ exists for all $t\ge 0$, remains in $D$, and satisfies $x(t;x_0)\to 0$ as $t\to\infty$. Then there is an open neighbourhood $U\subset D$ of $0$ and a function $V\in C^1(U)$ such that $V$ is positive definite on $U$ and

\begin{align*} \nabla V(x)\cdot f(x)<0 \qquad \text{for all } x\in U\setminus\{0\}. \end{align*}

Thus $V$ is a strict Lyapunov function for the equilibrium on $U$.