Backbones in the parameter plane of the Hénon map

Parameter plane (b, a) of the real Henon map has been investigated for curves of bifurcation, curves of homoclinic heteroclinic onsets, and also searching for borders of areas variously characterized. Such curves are, in general, complicated and show singularities. Pieces of two monotone curves, spanning the (b, a) parameter plane of the real Henon map, can be detected in four quite different studies appeared along the years 1982–2008. We study the extent of their similarity to read and interpret them into the same curves. To us, these two curves are the accumulation loci of bifurcation curves of two principal families of periodic sinks of type “period-adding machine.” We call them “backbones,” because they are monotone; moreover, they are the borders of some important regions in the (b, a)-plane. Hamouly and Mira in 1982 [C. R. Acad. Sc. Paris1 293, 525–528 (1982)] studied the structure of bifurcation of periodic orbits and their mutual position and intersection. Gonchenko et al. [SIAM J. Appl. Dyn. Syst...