# 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.

## Proof

## Source of Name

This entry was named for Ivar Otto Bendixson and Henri Claudius Rosaris Dulac.