Christophe LETELLIER
25/11/2012

**The Hénon map**

**Michel Hénon**

The Hénon map was discovered by Michel Hénon from the Nice Observatory in 1976 [1]. This is a two-dimensional discrete map reading as

When it is plotted *x _{n}* versus

**Fig. 1. Chaotic attractor solution to the Hénon map.**

**A suspension of the Hénon map**

Starting from the point of view that a discrete map can always be viewed as a first-return map to a Poincaré section of a flow, John Starrett & Craig Nicholas proposed a *suspension* of the Hénon map, that is, a three-dimensional flow whose first-return map to a Poincaré section is equivalent to the Hénon map [2]. They used the orientation preserving map (quite similar to the original Hénon map)

with *a*=4/3 and *b*=-1 (Fig. 2).

**Fig. 2. Chaotic attractor solution to the orientation preserving Hénon map investigated by Starrett & Nicholas.**

They then extracted three unstable periodic orbits from the chaotic attractor, constructed a suspension from these orbits using simple geometric transformations (a contraction, a bend, and a shift). They thus used the three-dimensional representation of these orbits to obtain a global model (using a projection on three multivariate polynomial functions) whose equations are provided in [2]. A numerical simulation of the global model obtained and a computation of its first-return map to a Poincaré section are shown in Fig. 3. Compare the return map to the Hénon map they started from.

**Fig. 3. Suspension of the orientation preserving Hénon map and a first-return map to its Poincaré section.**

[1] **M. Hénon**, A two-dimensional mapping with a strange attractor, *Communications in Mathematical Physics*, **50**, 69-77, 1976.

[2] **J. Starrett & C. Nicholas**, A suspension of the Hénon map by periodic orbits, *Chaos, Solitons & Fractals*, **45**, 1486-1493, 2012.