# Bendixson–Dulac theorem

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

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

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