We study the qualitative dynamics of two-parameter families of planar maps of the form F^e(x, y) = (y, -ex+f(y)), where f :R -> R is a C
3
map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f(y) = u —y
2
or (in different coordinates) f(y) = Ay(1 —y), in which case F^
e
is the Henon map. The maps F
e
have constant Jacobian determinant
e
and, as
e
-> 0, collapse to the family f^. 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/
ue
, for small
e
. Moreover, we are able to extend these results to the area preserving family F/u.
1
, thereby obtaining (partial) bifurcation sets in the (/u, e)-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 (
/u, e
) -parameter plane between
e
= 0 and
e
= 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F
/u.e
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.
A Chaotic Confusion-diffusion Image Encryption Based on Henon Map
