chapter  5
18 Pages

The He´non Model

The He´non model is a two-dimensional model introduced and analysed by He´non

(1976). This model can also be expressed by a delay difference equation:

xt+1 = bxt−1 + 1 − ax2t (5.1) By introducing a new variable, yt = xt+1, (5.1) is transformed into a system of two

ordinary difference equations:

xt+1 = yt + 1 − ax2t yt+1 = bxt


The Jacobian determinant of (5.2) is given by:

det J =

∣∣∣∣∣∣∣ ∂xt+1 ∂xt

∣∣∣∣∣∣∣ = −b When b = 1, the resulting map is area preserving, but the system is unstable. The

system becomes stable for 0 < b < 1. A variety of maps appear when a and b vary

over a range of values, and the resulting maps take the form of chaotic attractors.

We will see a variety of these attractors in this chapter. The most known chaotic

attractor of the He´non model appears when a = 1.4 and b = 0.3, and is illustrated in

Figure 5.1.