Bendixson–Dulac theorem
In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a function (called the Dulac function) such that the expression
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Dulac.svg/400px-Dulac.svg.png)
has the same sign () almost everywhere in a simply connected region of the plane, then the plane autonomous system
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 such that
in simply connected region . Let be a closed trajectory of the plane autonomous system in . Let be the interior of . Then by Green's theorem,
Because of the constant sign, the left-hand integral in the previous line must evaluate to a positive number. But on , and , 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 .
References[]
Henri Dulac (1870-1955) was a French mathematician from Fayence
- ^ Burton, Theodore Allen (2005). Volterra Integral and Differential Equations. Elsevier. p. 318. ISBN 9780444517869.
- Differential equations
- Theorems in dynamical systems