The Poincaré-Bendixson theorem on the Klein bottle for continuous vector fields (Q841388)

Let \(X\) be a continuous vector field defined on the Klein bottle. As usual \(\omega(\gamma)\) denotes the omega limit set of a trajectory \(\gamma\). It is said that \(\gamma\) is weakly \(\omega\)-recurrent if there exists \(p\in\gamma\) such that \(p\in\omega(\gamma)\). When \(\gamma\subset\omega(\gamma) \) we say that \(\gamma\) is \(\omega\)-recurrent. The authors prove the following Poincaré-Bendixson result: Assume that \(X\) has finitely many singularities. Then: (i) if \(\gamma\) be a weakly \(\omega\)-recurrent injective trajectory then \(\omega(\gamma)\) is either a periodic orbit or a graph; (ii) if \(\gamma\) is an injective trajectory which is not weakly \(\omega\)-recurrent, then \(\omega(\gamma)\) is either a singularity, or a periodic orbit, or a graph. As a consequence \(X\) has not \(\omega\)-recurrent injective trajectories.