Generalized Hénon Map and Bifurcations of Homoclinic Tangencies

We study two-parameter bifurcation diagrams of a generalized Henon map (GHM) that is known to describe dynamics of iterated maps near homoclinic and heteroclinic tangencies. We prove the nondegeneracy of codimension (codim) 2 bifurcations of fixed points of the GHM analytically and compute its various global and local bifurcation curves numerically. Special attention is given to the interpretation of the results and their application to the analysis of bifurcations of the homoclinic tangency of a neutral saddle in two-parameter families of planar diffeomorphisms. In particular, an infinite cascade of homoclinic tangencies of neutral saddle cycles is shown to exist near the homoclinic tangency of the primary neutral saddle.