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.