Integral invariants and limit sets of planar vector fields |
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Authors: | Isaac A. Garcí a,Douglas S. Shafer |
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Affiliation: | a Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Spain b Mathematics Department, University of North Carolina at Charlotte, Charlotte, NC 28223, USA |
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Abstract: | We show that if a planar system of differential equations admits an inverse integrating factor V defined in a neighborhood of a singular point with exactly one zero eigenvalue then V vanishes along any separatrix of the singular point. Additionally we prove that if K is a compact α- or ω-limit set that contains a regular point (or has an elliptic or parabolic sector if not), and if V is defined on a neighborhood of K, then V vanishes at at least one point of K (and on all of K if V is real analytic or Morse). |
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