Hénon Map

There are at least two maps known as the Hénon map.

The first is the two-dimensional dissipative quadratic map given by the coupled equations


(Hénon 1976).


The strange attractor illustrated above is obtained for alpha=1.4 and beta=0.3.


The illustration above shows two regions of space for the map with alpha=0.2 and beta=1.01 colored according to the number of iterations required to escape (Michelitsch and Rössler 1989).


The plots above show evolution of the point (0,0) for parameters (alpha,beta)=(0.2,0.9991) (left) and (0.2,-0.9999) (right).

The Hénon map has correlation exponent 1.25+/-0.02 (Grassberger and Procaccia 1983) and capacity dimension 1.261+/-0.003 (Russell et al. 1980). Hitzl and Zele (1985) give conditions for the existence of periods 1 to 6.

A second Hénon map is the quadratic area-preserving map


(Hénon 1969), which is one of the simplest two-dimensional invertible maps.

See also

Bogdanov Map, Gingerbreadman Map, Lozi Map, Quadratic Map

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Dickau, R. M. "The Hénon Attractor.", 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.Hénon, M. "Numerical Study of Quadratic Area-Preserving Mappings." Quart. Appl. Math. 27, 291-312, 1969.Hénon, M. "A Two-Dimensional Mapping with a Strange Attractor." Comm. Math. Phys. 50, 69-77, 1976.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.Michelitsch, M. and Rössler, O. E. "A New Feature in Hénon's Map." Comput. & Graphics 13, 263-275, 1989. Reprinted in Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (Ed. C. A. Pickover). Amsterdam, Netherlands: Elsevier, pp. 69-71, 1998.Morosawa, S.; Nishimura, Y.; Taniguchi, M.; and Ueda, T. "Dynamics of Generalized Hénon Maps." Ch. 7 in Holomorphic Dynamics. Cambridge, England: Cambridge University Press, pp. 225-262, 2000.Peitgen, H.-O. and Richter, D. H. The Beauty of Fractals: Images of Complex Dynamical Systems. New York: Springer-Verlag, 1986.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.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 95-97, 1991.

Referenced on Wolfram|Alpha

Hénon Map

Cite this as:

Weisstein, Eric W. "Hénon Map." From MathWorld--A Wolfram Web Resource.

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