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. "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,

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

$=\oint _{C}^{}{\varphi \left(-{\dot {y}}dx+{\dot {x}}dy\right)}.$

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

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