  Stickiness is a well known phenomenon in which chaotic orbits expend an
expressive amount of time in specific regions of the chaotic sea. This
phenomenon becomes important when dealing with area-preserving open systems
because, in this case, it leads to a temporary trapping of orbits in certain
regions of phase space. In this work, we propose that the different scenarios
of dynamical trapping can be explained by analyzing the crossings between
invariant manifolds. In order to corroborate this assertion, we use an adaptive
refinement procedure to approximately obtain the sets of homoclinic and
heteroclinic intersections for the area-preserving H\'enon map, an archetype of
open systems, for a generic parameter interval. We show that these sets have
very different statistical properties when the system is highly influenced by
dynamical trapping, whereas they present similar properties when stickiness is
almost absent. We explain these different scenarios by taking into
consideration various effects that occur simultaneously in the system, all of
which are connected with the topology of the invariant manifolds.

Chaotic Dynamics;Dynamical Systems;Computational Physics