Analytical hierarchy: Difference between revisions
en>Cydebot m Robot - Moving category Recursion theory to Computability theory per CFD at Wikipedia:Categories for discussion/Log/2011 February 5. |
en>Trovatore one link is fine; two in a row is excessive, and piping wikilinks to math markup is usually a bad idea |
||
| Line 1: | Line 1: | ||
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]] | |||
: <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}} | |||
Revision as of 21:21, 3 August 2013
In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a function (called the Dulac function) such that the expression
has the same sign () almost everywhere in a simply connected region of the plane, then the plane autonomous system
has no periodic solutions lying entirely within the region.[1] "Almost everywhere" means everywhere except possibly in a set of 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 such that
in simply connected region . Let be a closed trajectory of the plane autonomous system in . Let be the interior of . Then by Green's Theorem,
But on , and , so the integral evaluates to 0. This is a contradiction, so there can be no such closed trajectory .
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro. Template:Applied-math-stub
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534