= Solution
The <Bendixson-Dulac criterion> says the following. Let $D\subseteq\mathbb R^2$ be a <simply connected domain>, let $F=(f,g)$ be continuously differentiable, and suppose there is a continuously differentiable function $B$ such that
$$
\nabla\mathbin{\cdot}(BF)
=\frac{\partial(Bf)}{\partial x}
+\frac{\partial(Bg)}{\partial y}
$$
has one sign throughout $D$ and does not vanish identically on any open subset. Then $\dot z=F(z)$ has no <periodic orbit> contained in $D$.
Indeed, if a periodic orbit $\Gamma$ existed, simple connectivity would put its interior $U$ inside $D$. The vector field $F$, and hence $BF$, is tangent to $\Gamma$, so its outward flux is zero. The <divergence theorem> would give
$$
0=\int_\Gamma BF\mathbin{\cdot}n\,ds
=\iint_U\nabla\mathbin{\cdot}(BF)\,dx\,dy,
$$
contradicting the sign hypothesis.
The <Poincaré-Bendixson theorem> states that a nonempty compact <omega-limit set> of a continuously differentiable planar flow that contains no <equilibrium point> is a periodic orbit. More generally, when the omega-limit set contains only finitely many equilibria, it is either an equilibrium, a periodic orbit, or a union of equilibria and connecting trajectories.
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