Bendixson–Dulac theorem

In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a

(called the Dulac function) such that the expression 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 1923 using Green's theorem.

be a closed trajectory of the plane autonomous system in

Then by Green's theorem, Because of the constant sign, the left-hand integral in the previous line must evaluate to a positive number.

, so the bottom integrand is in fact 0 everywhere and for this reason the right-hand integral evaluates to 0.

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