Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region - Pouria Ramazi; Ming Cao; Jacquelien Scherpen (Electronic Journal of Qualitative Theory of Differential Equations)
This paper tightens the classical Poincaré–Bendixson theory for a positively invariant, simply-connected compact set $\mathcal M$ in a continuously differentiable planar vector field by further characterizing for any point $p\in \mathcal M$, the composition of the limit sets $ω (p)$ and $α(p)$ after counting separately the fixed points on $\mathcal M$'s boundary and interior. In particular, when $\mathcal M$ contains finitely many boundary but no interior fixed points, $ω (p)$ contains only a single fixed point, and when $\mathcal M$ may have infinitely many boundary but no interior fixed points, $ω (p)$ can, in addition, be a continuum of fixed points. When $\mathcal M$ 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 $\mathcal M$ 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.