Bendixson-Dulac Theorem

Theorem
Suppose there exists a continuously differentiable function $\alpha(x,y)$ on a simply connected domain.

Suppose that:
 * $\nabla \cdot (\alpha F)$

is either always positive or always negative:

Then the two-dimensional autonomous system:
 * $(x,y)' = F(x,y)$

does not have a periodic solution.