TITLE: A NUMERICAL EXPLORATION OF WEAKLY DISSIPATIVE TWO-DIMENSIONAL MAPS AUTHORS: Carles Sim\'o and Arturo Vieiro Dept de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona Gran Via, 585, 08007 Barcelona, Spain E-mail: carles@maia.ub.es, vieiro@maia.ub.es ABSTRACT: The aim of this work is to study the global dynamics of a planar weakly dissipative map around a perturbation of an elliptic fixed point. If the dissipative perturbation is assumed to be radial the map presents different domains depending on the topological behaviour. When showing these main domains the different $\omega$-limits that can co-exist are also described. Furthermore, the parametric study of the mechanism of destruction of resonances and the evolution of invariant objects of the phase space are displayed. On the other hand, the probability of capture in different kinds of resonances as a function of the parameters of the map and the dissipation parameter is also given. The numerical approach to these ideas provides a first step to develop a global theory for this kind of maps. In order to exhibit the generic properties of weakly dissipative maps a radially dissipative version of the H\'enon map is considered.