Families of periodic sinks: the quasi-conservative case

We will discuss, as a classical example of dissipative map, the H\'enon map when the dissipation vanishes. Using a numerical continuation that we devised, and called ``dribbling method'' \cite{FT1}, one can follow bifurcation paths from the highly degenerate area-preserving case into the dissipative one, organizing families of coexisting attractive periodic orbits with diverging period. The coexistence of sinks is greater and greater approaching the conservative case. When the dissipation parameter goes to zero, we will give numerical evidence of the coexistence of such periodic orbits, in the coordinate and parameter space values . We discuss the dependence of the stability range of periodic orbits with respect to the period. As the period $p$ diverges, we describe here the renormalization scheme we set up to study the families. The families we study all appear as homoclinic bifurcation, and the fixed point causing the homoclinic onset also structures the renormalization scheme. Using the same dribbling method, as further promising application, we also deal with the dissipative Standard map