 Hénon map

The Hénon map is a two-parameter quadratic planar map, conventionally $(X,Y)\mapsto (1-a{X}^2+Y,b_{H}X)$. The alternative family $(x,y)\mapsto (y,\mu -bx-y^2)$ has constant Jacobian determinant $b$. When $\mu \ne 0$, the change $X=y/\mu$, $Y=-bx/\mu$ gives the conventional parameters $a=\mu$, $b_H=-b$. Fixed points in the alternative normalization satisfy $x=y$ and $y^2+(1+b)y-\mu =0$. Its invertibility requires $b\ne 0$; at $b=1$ it preserves area.

The Hénon map is a discrete-time dynamical system that is commonly studied in the field of chaos theory. It is a simple, quadratic map that can exhibit chaotic behavior, making it an important example in the study of dynamical systems. The map is named after the French mathematician Michel Hénon, who introduced it in the context of studying the dynamics of celestial mechanics and later generalized it for various applications.