Henon map is a well-studied classical example of area-contracting maps, modelling dissipative dynamics. The rich phenomena of coexistence of stable islands and their separatrices is typical of area-preserving maps, modelling conservative dynamics. In this paper we use the Henon map to ascertain that coexistence of sinks is greater and greater approaching the conservative case, and that part of it can be organized following a renormalization argument. Using a numerical continuation that we devised, and called "dribbling method" , one can follow bifurcation paths from the area-preserving case into the dissipative one, organizing families of coexisting attractive periodic orbits with diverging period. When the dissipation parameter goes to zero, we will give numerical evidence of the increasing coexistence of such periodic orbits, in the coordinate and parameter space values. Vanishing dissipation and diverging period constitute a double limit that we study as such, giving evidence of a singularity in the limit. The families we study all appear as homoclinic bifurcation, and the fixed point causing the homoclinic onset also structures the renormalization scheme. One of the goals of this paper is to improve the results obtained by looking to higher periods, and to approach dissipation down to an area-contraction factor of 1 - 10(-8) Using the same dribbling method, as further promising application, we also deal with the dissipative Standard map.
Falcolini, C., Tedeschini-Lalli, L. (2018). Diverging period and vanishing dissipation: Families of periodic sinks in the quasi-conservative case. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 38(12), 6105-6122 [10.3934/dcds.2018263].