Bendixson–Dulac theorem

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In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a function (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 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[edit]

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,

But on , and , so the integral evaluates to 0. This is a contradiction, so there can be no such closed trajectory .

References[edit]

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

  1. ^ Burton, Theodore Allen (2005). Volterra Integral and Differential Equations. Elsevier. p. 318. ISBN 9780444517869.