  We consider a family of planar systems introduced in 1991 by Benedicks and
Carleson as a toy model for the dynamics of the so-called H\'enon maps. We show
that Smale's Axiom A property is $C^1$-dense among the systems in this family,
despite the existence of $C^2$-open subsets (closely related to the so-called
Newhouse phenomena) where Smale's Axiom A is violated. In particular, this
provides some evidence towards Smale's conjecture that Axiom A is a $C^1$-dense
property among surface diffeomorphisms.
  The basic tools in the proof of this result are: 1) a recent theorem of
Moreira saying that stable intersections of dynamical Cantor sets (one of the
main obstructions to Axiom A property for surface diffeomorphisms) can be
destroyed by $C^1$-perturbations; 2) the good geometry of the dynamical
critical set (in the sense of Rodriguez-Hertz and Pujals) thanks to the
particular form of Benedicks-Carleson toy models.
