Backbones in the Parameter Plane of the Hénon Map

Parameter plane (b, a) of the real Hénon 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 Hénon 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. 4, 407–436 (2005)] display the continuation (in parameter plane) of the first heteroclinic connection and of the first homoclinic connection between the two fixed points of the map. Alligood and Sauer [Commun. Math. Phys. 120, 105–119 (1988)] studied parameter regions characterized by the same rotation number of the “accessible” periodic saddle. Finally, Lorenz [Physica D 237, 1689–1704 (2008)] in 2008 draws areas in the parameter plane statistically characterized by a finite attractor. In this paper, we show how these criteria interact. We therefore conjecture that the wealth of curves of homoclinic onsets could be in general hierarchized by the structure of accessible saddles.