Bifurcations of One- and Two-Dimensional Maps

We study the qualitative dynamics of two-parameter families of planar maps of the form F_μ,ɛ(x,y)=(y,-ɛ x+f_μ(y)), where [Note: See the image of page 43 for this formatted text] f_μ: R --> R is a C³ map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f_μ(y) = μ -y² or (in different coordinates) f_λ(y) = λ y(1-y), in which case F_μ,ɛ is the Henon map. The maps F_μ,ɛ have constant Jacobian determinant ɛ and, as ɛ --> 0, collapse to the family f_μ. The behaviour of such one-dimensional families is quite well understood, and we are able to use their bifurcation structures and information on their non-wandering sets to obtain results on both local and global bifurcations of F_μ,ɛ, for small ɛ . Moreover, we are able to extend these results to the area preserving family F_μ.1, thereby obtaining (partial) bifurcation sets in the (μ,ɛ )-plane. Among our conclusions we find that the bifurcation sequence for periodic orbits, which is restricted by Sarkovskii's theorem and the kneading theory for one-dimensional maps, is quite different for two-dimensional families. In particular, certain periodic orbits that appear at the end of the one-dimensional sequence appear at the beginning of the area preserving sequence, and infinitely many families of saddle node and period doubling bifurcation curves cross each other in the (μ,ɛ )-parameter plane between ɛ = 0 and ɛ = 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F_μ,ɛ and of the associated sequences of periodic bifurcations that accumulate on them. We illustrate our results with some numerical computations for the orientation-preserving Henon map.