Proof of Dulac’s Criteria

Consider the the planar system x˙=f(x), where f=(X,Y)t and x=(x,y)t. Consider the vector field (-ρY,ρX). Suppose that there is a periodic orbit contained in E associated to the planar system. Let γ be that periodic orbit. We have:


On the other hand, the region within E that is limited by γ is simply connected because E is simply connected. Let E~ be the region limited by γ. Then, by Green’s theorem, we have:


Because E~ has positive area and the integrand function has constant signal, then this integral is different from zero. This is a contradictionMathworldPlanetmathPlanetmath. So there are no periodic orbits. \qed

Title Proof of Dulac’s Criteria
Canonical name ProofOfDulacsCriteria
Date of creation 2013-03-11 19:17:10
Last modified on 2013-03-11 19:17:10
Owner Filipe (28191)
Last modified by (0)
Numerical id 9
Author Filipe (0)
Entry type Proof
Classification msc 34C25
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