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

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.