Hénon map
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Image:HenonMapImage.png 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 parameters, a and b, which for the canonical Hénon map have values of a = 1.4 and b = 0.3. For the canonical 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 type of behavior of the map at different parameter values may be obtained from its bifurcation diagram.
The map was introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. For the canonical map, 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) for the attractor of the canonical map.
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
- M. Hénon (1976). A two-dimensional mapping with a strange attractor. Communications of Mathematical Physics 50: 69-77.
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- D.A. Russel, J.D. Hanson, and E. Ott (1980). Dimension of strange attractors. Phys. Rev. 45: 1175. (LINK)
- P. Grassberger and I. Procaccia (1983). Measuring the strangeness of strange attractors. Physica 9D: 189-208. (LINK)
