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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>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
- Template:Journal reference
- Template:Book reference
- Template:Journal reference (LINK) (http://prola.aps.org/abstract/PRL/v45/i14/p1175_1)
- Template:Journal reference (LINK) (http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1983PhyD....9..189G&db_key=PHY)