Abstract

Global behavior of continuous and discrete dynamical systems is an interesting question. For local studies, Poincaré-Bendixon Theorem is a very powerful tool for continuous dynamical systems defined in $R2$, but in the case of discrete dynamical system is not so. It works if, for instance, we consider the time T-map of a flow given by a vector field. But what happens if the map is not even differentiable in $R2$? In this case chaotic behavior can appear.

First, we will see the relevance of Brower′s Lemma of Translation Arcs and Degree Theory in describing the local dynamics of a fixed point and we will give some global dynamic configurations of orientation preserving homeomorphisms defined in $R2$.

Secondly, we will prove the existence of a global asymptotically attacting fixed point for continuous and injective maps of the plane (not necessarily homeomorphisms).

Finally we will see the relationship between these results and Markus-Yamabe Conjecture in dimension two. Moreover, we will give some applications to Theory of Oscillations and Mathematical Biology via Poincaré map.