The main purpose of this paper is to investigate dynamical systems
$${F : \mathbb{R}^2 \rightarrow \mathbb{R}^2}$$
of the form F(x, y) = (f(x, y), x). We assume that
$${f : \mathbb{R}^2 \rightarrow \mathbb{R}}$$
is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x0 = (x0, y0), such that the orbit
$$ O({\rm{x}}_0) = \{{\rm{x}}_0, {\rm{x}}_1 = F({\rm{x}}_0), {\rm{x}}_2 = F({\rm{x}}_1), . . . \}, $$
is bounded, O(x0) converges provided that the set of fixed point of F is totally disconnected and F does not admit periodic orbits of prime period two. The obtained result is used to show that all aperiodic orbits can be removed from the dynamics of the map H of Hénon. The goal can be achieved by perturbing H so that the perturbed map H1 does not have any periodic point of prime period two.