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

According to Dulac theorem any 2D autonomous system with a periodic orbit has a region with positive and a region with negative divergence inside such orbit. Here represented by red and green regions respectively

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.

Proof

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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  .

See also

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  • Limit cycle § Finding limit cycles
  • Liouville's theorem (Hamiltonian), similar theorem with  

References

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