Let $f \in C^1(E)$ where E is an open subset of $\mathbb{R^n}$ containing a periodic orbit $\gamma(t)$ of $x'=f(x)$ of period $T$. Then $\gamma(t)$ is not asymptotically stable unless $$\int_0^T \nabla \cdot f(\gamma(t))dt \leq 0$$
Comments: Tried using the Lioville's Theorem using the fact that after the linearization of $x'=f(x)$ for $x'=A(t)x$, where $A(t)= Df(\gamma(t)).$