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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


:<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]]
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: <math>\frac{ dx }{ dt } = f(x,y),</math>
 
: <math>\frac{ dy }{ dt } = g(x,y)</math>
 
has no [[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 1933 using [[Green's theorem]].
 
==Proof==
Without loss of generality, let there exist a function <math> \varphi(x, y)</math> such that
 
:<math>\frac { \partial (\varphi f) }{ \partial x } +\frac { \partial (\varphi g) }{ \partial y } >0</math>
 
in simply connected region <math>R</math>. Let <math>C</math> be a closed trajectory of the plane autonomous system in <math>R</math>. Let <math>D</math> be the interior of <math>C</math>. Then by [[Green's Theorem]],
 
:<math>\iint _{ D }^{  }{ \left( \frac { \partial (\varphi f) }{ \partial x } +\frac { \partial (\varphi g) }{ \partial y }  \right) dxdy } =\oint _{ C }^{  }{ -\varphi gdx+\varphi fdy } </math>
 
:<math>=\oint _{ C }^{  }{ \varphi \left( -\dot { y } dx+\dot { x } dy \right)  }. </math>
 
But on <math>C</math>, <math>dx=\dot { x } dt</math> and <math>dy=\dot { y } dt</math>, so the integral evaluates to 0. This is a contradiction, so there can be no such closed trajectory <math>C</math>.
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Bendixson-Dulac Theorem}}
[[Category:Differential equations]]
[[Category:Theorems in dynamical systems]]
{{Applied-math-stub}}

Latest revision as of 14:01, 21 December 2014


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