Hénon map

In mathematics, the Hénon map is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point $(x n, y n)$ in the plane and maps it to a new point: ${\begin{cases}x_{n+1}=1-ax_{n}^{2}+y_{n}\\y_{n+1}=bx_{n}\end{cases}}$ The map depends on two parameters, $a$ and $b$ , which for the classical Hénon map have values of $a = 1.4$ and $b = 0.3$ . For the classical values, the Hénon map is chaotic. For other values of $a$ and $b$ , the map may be chaotic, intermittent, or converge to a periodic orbit. An overview of the map's behavior at different parameter values can be seen in its orbit diagram.

The map was introduced by Michel Hénon as a simplified model for the Poincaré section of the Lorenz system. For the classical map, an initial point in the plane will either approach a set of points known as the Hénon strange attractor, or it will diverge to infinity. The Hénon attractor is a fractal, smooth in one direction and a Cantor set in another. Numerical estimates for the fractal dimension of the strange attractor for the classical map yield a correlation dimension of 1.21 ± 0.01 and a box-counting dimension of 1.261 ± 0.003.