Editing Bendixson–Dulac theorem

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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]].

==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>
\begin{align}
& \iint_D \left( \frac { \partial (\varphi f) }{ \partial x } +\frac { \partial (\varphi g) }{ \partial y }  \right) \,dx\,dy 
=\iint_D \left( \frac { \partial (\varphi \dot { x }) }{ \partial x } +\frac { \partial (\varphi \dot { y }) }{ \partial y }  \right) \,dx\,dy \\[6pt]
= {} & \oint_C \varphi \left( -\dot { y } \,dx+\dot { x } \,dy\right)
=\oint_C \varphi \left( -\dot { y }\dot { x }+\dot { x }\dot { y }\right)\,dt=0
\end{align}
</math>

Because of the constant sign, the left-hand integral in the previous line must evaluate to a positive number. But on <math>C</math>, <math>dx=\dot { x } \,dt</math> and <math>dy=\dot { y } \,dt</math>, so the bottom integrand is in fact 0 everywhere and for this reason the right-hand integral evaluates to&nbsp;0. This is a contradiction, so there can be no such closed trajectory <math>C</math>.

== See also ==
* {{slink|Limit_cycle#Finding limit cycles}}
* [[Liouville's theorem (Hamiltonian)]], similar theorem with <math>\frac{ dq }{ dt } =\frac{\partial H(q,p)}{\partial p}\, (=f(q,p)), \frac{ dp }{ dt } =-\frac{\partial H(q,p)}{\partial q}\, (=g(q,p))</math>

==References==
{{reflist}}

{{DEFAULTSORT:Bendixson-Dulac Theorem}}
[[Category:Differential equations]]
[[Category:Theorems in dynamical systems]]


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