The Dynamics of the Henon Map

Abstract

The purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the Henon map with expansion combined with strong contraction. Our study is modeled on the treatment of the one-dimensional system x→1-ax 2 and we study the perturbation of a from the value a=2 and b small. A surprising number of complications arise when b>0. The computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up. The central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the Henon one