= Solution
Let $\dot z=F(z)$ be a $C^1$ planar system on a simply connected domain $D$. The <Bendixson-Dulac criterion> states that if some $B\in C^1(D)$ makes
$$
\nabla\mathbin{\cdot}(BF)
$$
have one sign throughout $D$ and not vanish identically on any open subset, then $D$ contains no periodic orbit.
Indeed, if a periodic orbit $\Gamma$ bounded a region $\Omega\subset D$, then $F$ and hence $BF$ would be tangent to $\Gamma$. Its normal flux would therefore vanish. The divergence theorem would give
$$
0=\int_\Gamma BF\mathbin{\cdot}n\,ds
=\iint_\Omega\nabla\mathbin{\cdot}(BF)\,dA,
$$
contradicting the sign hypothesis. Taking $B=1$ gives the usual Bendixson divergence criterion.
The stability version of the divergence test concerns an existing periodic orbit $\gamma$ of period $T$. Liouville's formula for the variational equation shows that its nontrivial <Floquet multiplier> is
$$
\rho=\exp\left(
\int_0^T\nabla\mathbin{\cdot}F(\gamma(t))\,dt
\right).
$$
\b[Thus the <divergence test for a planar periodic orbit> says that $\gamma$ is asymptotically stable if the integral is negative and unstable if it is positive.]
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