Bendixson-Dulac Theorem

Theorem

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

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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

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Source of Name

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