The dynamics and the periodic orbits of the Henon map

Ross, David J.
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University of Guelph

The Henon map (H) is a mapping from R\sp2 to R\sp2 with two associated parameters a and b. The map H is defined by:$$H\sb{a,b}{x\choose y}={a-by-x\sp2\choose x}$$The parameters are both real and there is only one non-linear term and it is noted that the complicated dynamics which result from iteration under H are caused by one of the simplest non-linear two-dimensional maps. This is analogous to the way that the quadratic family gives fantastically complicated dynamics in one dimension. In this thesis, we review the known theory of the Henon map which is a class of reversible quadratic maps from the plane to itself. We also present a new result concerning the birth of two period three orbits in a classical saddle node bifurcation. A corollary of this result demonstrates explicitly the different nature of maps from the plane to itself from maps of the line to itself. We also discuss the extensive numerical work which was done concerning a saddle node bifurcation which gives us two period 4 orbits.

Henon map, dynamics, periodic orbits, reversible quadratic maps, saddle node bifurcation