Bendixson-Dulac criterion

If $D$ is simply connected and a $C^1$ function $B$ makes $\nabla \cdot (BF)$ have one sign and not vanish identically on any open subset of $D$, then the planar system $\dot{x}=F(x)$ has no periodic orbit lying in $D$.

A multiplier $B(x,y)$ whose rescaled planar vector field has divergence of one strict sign can exclude periodic orbits in a simply connected region. For a system $\dot{x}=xF$, $\dot{y}=yG$ in the positive quadrant, $B=1/(xy)$ is often effective.