Editing Bendixson–Dulac theorem (section)

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In [[mathematics]], the '''Bendixson–Dulac theorem''' on [[dynamical system]]s states that if there exists a <math>C^1</math> [[function (mathematics)|function]] <math> \varphi(x, y)</math> (called the Dulac function) such that the expression

[[File:Dulac.svg|thumb|400px|right|
According to Dulac theorem any 2D autonomous system with a periodic orbit has a region with positive and a region with negative divergence inside such orbit. Here represented by red and green regions respectively]]

:<math>\frac{ \partial (\varphi f) }{ \partial x } + \frac{ \partial (\varphi g) }{ \partial y }</math>

has the same sign (<math>\neq 0</math>) [[almost everywhere]] in a [[simply connected]] region of the plane, then the [[plane autonomous system]]

: <math>\frac{ dx }{ dt } = f(x,y),</math>

: <math>\frac{ dy }{ dt } = g(x,y)</math>

has no nonconstant [[periodic solution]]s lying entirely within the region.<ref name=Burton2005>{{cite book|last=Burton|first=Theodore Allen|title=Volterra Integral and Differential Equations|year=2005|publisher=Elsevier|isbn=9780444517869|page=318}}</ref> "Almost everywhere" means everywhere except possibly in a set of [[measure (mathematics)|measure]] 0, such as a point or line.

The theorem was first established by Swedish mathematician [[Ivar Bendixson]] in 1901 and further refined by French mathematician [[Henri Dulac]] in 1923 using [[Green's theorem]].