Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region

• Published in: Electronic journal of qualitative theory of differential equations Vol. 2024; no. 29; pp. 1 - 9
• Main Authors: Ramazi, Pouria, Cao, Ming, Scherpen, Jacquelien M. A.
• Format: Journal Article
• Language: English
• Published: University of Szeged 01.01.2024
• Subjects: limit set; planar vector field; poincaré–bendixson theory
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2024.1.29

This paper tightens the classical Poincaré–Bendixson theory for a positively invariant, simply-connected compact set M in a continuously differentiable planar vector field by further characterizing for any point p ∈ M , the composition of the limit sets ω ( p ) and α ( p ) after counting separately the fixed points on M 's boundary and interior. In particular, when M contains finitely many boundary but no interior fixed points, ω ( p ) contains only a single fixed point, and when M may have infinitely many boundary but no interior fixed points, ω ( p ) can, in addition, be a continuum of fixed points. When M contains only one interior and finitely many boundary fixed points, ω ( p ) or α ( p ) contains exclusively a fixed point, a closed orbit or the union of the interior fixed point and homoclinic orbits joining it to itself. When M contains in general a finite number of fixed points and neither ω ( p ) nor α ( p ) is a closed orbit or contains just a fixed point, at least one of ω ( p ) and α ( p ) excludes all boundary fixed points and consists only of a number of the interior fixed points and orbits connecting them.