Experimentarium DigitaleBy playing with the parameters of the Lorenz equations and using a Poincaré section, Pomeau and Ibanez demonstrated the formation mechanism of a Smale horseshoe. Pomeau presented his work at a seminar attended by Michel Hénon who then conceived a very simple model of quadratic transformation of the plane which simulates, when a parameter varies, the mechanism of formation of a horseshoe: it is the famous Hénon model.
The model is defined as follows. Given $(x_0,y_0)$ in the place, one computes its orbit $(x_1,y_1),(x_2,y_2),...$ by successive iterations:
$$ \begin{cases} x_{n+1} =y_n+1-ax_n^2 \\ y_{n+1} =b x_n \end{cases} $$
where $a,b$ are parameters.
Numerical exploration of this model shows, for certain values of the parameters, the existence of a "strange attractor" which has a fractal structure. The values in Hénon’s paper are $a=1.4$, $b=0.3$.
The fact that this attractor really exists, and is not just a numerical belief, remained an open problem until the late 1980s. It was in 1991 that Benedicks and Carleson first showed rigorously the existence of these attractors. Their theorem is a mathematical tour-de-force but does not cover the values of the parameters $a=1.4$ et $b=0.3$.
In the following interactive experiment, we run 100 initial points at the same time to generate the attractor faster.
By clicking in the plane you can zoom in to see more details of the strange attractor, when it appears.
HTML code to integrate this simulation into your pages :
<iframe style="overflow: hidden;" src="/simulations/sysdyn-ModeleHenon-en/index.html" height="550" width="900" frameborder="0" scrolling="no"></iframe>