Bendixson's negative criterion

Hunting for limit cycles can be hard, so it is enormously useful to have a quick way to prove a region has none — to rule out periodic motion before you waste effort looking. Bendixson's negative criterion is exactly such a no-go test. With one easy computation it can certify that no closed orbit can fit inside a region, settling the question in the negative.

The criterion uses the divergence of the vector field. For a planar system x' = f(x,y), y' = g(x,y), form the divergence ∂f/∂x + ∂g/∂y — a measure of how much the flow expands or compresses area at each point. Bendixson's theorem says: if this divergence is not identically zero and never changes sign throughout a simply connected region (one with no holes), then that region contains no closed orbit. The reason is a clean area argument: around any closed orbit, the integral of the divergence over the enclosed area must be zero (the inflow and outflow balance), but a divergence of one fixed sign integrates to something nonzero — a contradiction, so no such orbit can exist.

Dulac's refinement multiplies the field by a cleverly chosen positive weight B(x,y) before taking the divergence, which often turns a sign-changing divergence into a one-signed one and rules out cycles where the raw criterion fails. The honest scope: this is a negative test only. It can prove no limit cycle exists in a region, but it can never prove one does — for that you turn to the Poincare-Bendixson theorem. And it applies only in simply connected regions; a hole in the region voids the area argument.