# Bendixson–Dulac theorem

In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a ${\displaystyle C^{1}}$ function ${\displaystyle \varphi (x,y)}$ (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
${\displaystyle {\frac {\partial (\varphi f)}{\partial x}}+{\frac {\partial (\varphi g)}{\partial y}}}$

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

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

has no nonconstant 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 ${\displaystyle \varphi (x,y)}$  such that

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

in simply connected region ${\displaystyle R}$ . Let ${\displaystyle C}$  be a closed trajectory of the plane autonomous system in ${\displaystyle R}$ . Let ${\displaystyle D}$  be the interior of ${\displaystyle C}$ . Then by Green's theorem,

{\displaystyle {\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 ${\displaystyle C}$ , ${\displaystyle dx={\dot {x}}\,dt}$  and ${\displaystyle dy={\dot {y}}\,dt}$ , 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 ${\displaystyle C}$ .

## References

Henri Dulac (1870-1955) was a French mathematician from Fayence

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