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

Abstract

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.