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HenonMapImage.png
Hénon attractor for a = 1.4 and b = 0.3

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 (xy) in the plane and maps it to a new point

<math>x_{n+1} = y_n+1-a x_n^2\,<math>
<math>y_{n+1} = b x_n\,<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 Michel 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. Numerical estimates yield a correlation dimension of 1.42 ± 0.02 (Grassberger, 1983) and a Hausdorff dimension of 1.261 ± 0.003 (Russel 1980) when a = 1.4 and b = 0.3.

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


Decomposition

The Hénon map may be decomposed into an area-preserving bend:

<math>(x_1,y_1) = (x, 1 - ax^2 + y)<math>

a contraction in the "x" direction:

<math>(x_2,y_2) = (bx_1, y_1)<math>

and a reflection in the line y = x:

<math>(x_3, y_3) = (y_2, x_2)<math>

References

See also

Template:Math-stub

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