# 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 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
${\frac {\partial (\varphi f)}{\partial x}}+{\frac {\partial (\varphi g)}{\partial y}}$ 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 nonconstant periodic solutions lying entirely within the region. "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,

{\begin{aligned}&\iint _{D}\left({\frac {\partial (\varphi f)}{\partial x}}+{\frac {\partial (\varphi g)}{\partial y}}\right)\,dx\,dy=\oint _{C}\left(-\varphi g\,dx+\varphi f\,dy\right)\\[6pt]={}&\oint _{C}\varphi \left(-{\dot {y}}\,dx+{\dot {x}}\,dy\right).\end{aligned}}

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