Bendixson-Dulac Theorem

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

Suppose that:
 * $\nabla \cdot \left({\alpha F}\right)$

is either always positive or always negative.

Then the two-dimensional autonomous system:
 * $ \left({x, y}\right)' = F \left({x, y}\right)$

does not have a periodic solution.