Bendixson–Dulac theorem

In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a function $\varphi(x, y)$ such that

$\frac{ \partial (\varphi f) }{ \partial x } %2B \frac{ \partial (\varphi g) }{ \partial y }$

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

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

has no periodic solutions. "Almost everywhere" means everywhere except possibly in a set of area 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.^[1]

References