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. "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 1923 using Green's theorem.