ABSTRACT
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
(5.2)
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.
