Bendixson-Dulac theorem

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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 } \ne 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.

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