# 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

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

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,

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

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

## References

Henri Dulac is a French mathematician (1870, 1955) from Fayence

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