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Dulac’s criteria


Let

˙𝐱=𝐟(𝐱)

be a planar system where 𝐟=(𝐗,𝐘)t and 𝐱=(x,y)t. Furthermore 𝐟∈C1(E) where E is a simply connected region of the plane. If there exists a function p(x,y)∈C1(E) such that βˆ‚(p(x,y)𝐗)βˆ‚x+βˆ‚(p(x,y)𝐘)βˆ‚y (the divergence of the vector field p(x,y)𝐟, βˆ‡β‹…p(x,y)𝐟) is always of the same sign but not identically zero then there are no periodic solution in the region E of the planar system. In addition, if A is an annular region contained in E on which the above condition is satisfied then there exists at most one periodic solution in A.

Title Dulac’s criteria
Canonical name DulacsCriteria
Date of creation 2013-03-22 13:37:09
Last modified on 2013-03-22 13:37:09
Owner Daume (40)
Last modified by Daume (40)
Numerical id 5
Author Daume (40)
Entry type Theorem
Classification msc 34C25