ITES2001 - Fourth Italian-Spanish Conference on
GENERAL TOPOLOGY AND ITS APPLICATIONS
Bressanone, 27-30 June 2001

On the Poincaré-Bendixson Theorem
Gabriel Soler López
Universidad de Cartagena, Spain

The abstract of the talk (see below) may be downloaded in one of the following formats:


The celebrated Poincar\'e-Bendixson Theorem assures that given a $C^1$-flow on
$\mathbb{S}^2$, $\Psi:\mathbb{R}\times\mathbb{S}^2\rightarrow\mathbb{S}^2$,
and a point $x$ whose $\omega$-limit set does not contains any singular point,
then $\omega_\Phi(x)$ is a closed orbit.  The aim of this talk is to show
in which phase spaces this Theorem also works, in particular we will see
that it is valid for the Klein Bottle and the Projective Plane with the
same formulation that in the case of $\mathbb{S}^2$.

In all the  compact and connected surfaces, except for the Torus, we will
see that the Theorem  works for flows of
$C^2$-class. Finally we will state the recent results in the Poincar\'e-Bendixson
Theory, namely we will state topological characterizations of the
$\omega$-limit sets in some phase spaces: Klein Bottle, Projective Plane and
$n$-dimensional sphere.



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