Bendixson-Dulac theorem

In mathematics, the Bendixson-Dulac theorem on dynamical systems states that if there exists φ("x","y")

:$$frac{ partial (phi f) }{ partial x } + frac{ partial (phi g) }{ partial y } e 0

"almost everywhere" in the region of interest, which must be simply connected, then the plane autonomous system

:$$frac{ dx }{ dt } = f(x,y),

:$$frac{ dy }{ dt } = g(x,y)

has no periodic solutions. "Almost everywhere" can mean everywhere except possibly a set of area 0, such as a point or line. This can be proved by Green's theorem.