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.