 Bendixson–Dulac theorem

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 $D$.

The Bendixson–Dulac theorem is a result in the field of dynamical systems and differential equations. It provides criteria to determine the existence of periodic orbits in a planar autonomous system of ordinary differential equations. Specifically, the theorem helps to identify non-existence of periodic orbits in a given planar system, which can be useful for analyzing the behavior of the system.