Bendixson–Dulac theorem

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In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a C^1 function  \varphi(x, y) (called the Dulac function) such that the expression

\frac{ \partial (\varphi f) }{ \partial x } + \frac{ \partial (\varphi g) }{ \partial y }

has the same sign (\neq 0) almost everywhere in a simply connected region of the plane, then the plane autonomous system

\frac{ dx }{ dt } = f(x,y),
\frac{ dy }{ dt } = g(x,y)

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[edit]

Without loss of generality, let there exist a function  \varphi(x, y) such that

\frac { \partial (\varphi f) }{ \partial x } +\frac { \partial (\varphi g) }{ \partial y } >0

in simply connected region R. Let C be a closed trajectory of the plane autonomous system in R. Let D be the interior of C. Then by Green's Theorem,

\iint _{ D }^{  }{ \left( \frac { \partial (\varphi f) }{ \partial x } +\frac { \partial (\varphi g) }{ \partial y }  \right) dxdy } =\oint _{ C }^{  }{ -\varphi gdx+\varphi fdy }
=\oint _{ C }^{  }{ \varphi \left( -\dot { y } dx+\dot { x } dy \right)  }.

But on C, dx=\dot { x } dt and dy=\dot { y } dt, so the integral evaluates to 0. This is a contradiction, so there can be no such closed trajectory C.

References[edit]

  1. ^ Burton, Theodore Allen (2005). Volterra Integral and Differential Equations. Elsevier. p. 318. ISBN 9780444517869.