Hénon Map

$\begin{figure}\begin{center}\BoxedEPSF{henon.epsf scaled 690}\end{center}\end{figure}$

A quadratic 2-D Map given by the equations

$\displaystyle x_{n+1}$	$\textstyle =$	$\displaystyle 1-\alpha {x_n}^2+y_n$	(1)
$\displaystyle y_{n+1}$	$\textstyle =$	$\displaystyle \beta x_n$	(2)

$\displaystyle x_{n+1}$	$\textstyle =$	$\displaystyle x_n\cos\alpha-(y_n-{x_n}^2)\sin\alpha$	(3)
$\displaystyle y_{n+1}$	$\textstyle =$	$\displaystyle x_n\sin\alpha+(y_n-{x_n}^2)\cos\alpha.$	(4)

The above map is for $\alpha=1.4$ and $\beta=0.3$ . The Hénon map has Correlation Exponent $1.25\pm 0.02$ (Grassberger and Procaccia 1983) and Capacity Dimension $1.261\pm 0.003$ (Russell et al. 1980). Hitzl and Zele (1985) give conditions for the existence of periods 1 to 6.

See also Bogdanov Map, Lozi Map, Quadratic Map

References

Dickau, R. M. ``The Hénon Attractor.'' http://forum.swarthmore.edu/advanced/robertd/henon.html.

Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 144-153, 1988.

Grassberger, P. and Procaccia, I. ``Measuring the Strangeness of Strange Attractors.'' Physica D 9, 189-208, 1983.

Hitzl, D. H. and Zele, F. ``An Exploration of the Hénon Quadratic Map.'' Physica D 14, 305-326, 1985.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 128-133, 1991.

Peitgen, H.-O. and Saupe, D. (Eds.). ``A Chaotic Set in the Plane.'' §3.2.2 in The Science of Fractal Images. New York: Springer-Verlag, pp. 146-148, 1988.

Russell, D. A.; Hanson, J. D.; and Ott, E. ``Dimension of Strange Attractors.'' Phys. Rev. Let. 45, 1175-1178, 1980.