Hénon map

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, y) in the plane and maps it to a new point

<math> f_{a,b}(x,y) = (y+1-a x^2, b x)<math>.

The map depends on two constants a and b, which have the canonical values of a = 1.4 and b = 0.3.

The map was introduced by Michele Hénon as a simplified model of the Poincaré section of the Lorenz model. For the canonical map (a = 1.4 and b = 0.3) an initial point of the plane will either approach a set of points known as the Hénon strange attractor, or diverge to infinity. The Hénon attractor is a fractal, smooth in one direction and a Cantor set in another.

As a dynamical system, the canonical Hénon map is interesting because, unlike the logistic map, its orbits defy a simple description.

See also Takens' theorem.

References

• M. Hénon (1976). "A two-dimensional mapping with a strange attractor". Communications of Mathematical Physics 50: 69–77.