The illustration above shows two regions of space for the map with and colored according to the number of iterations required
to escape (Michelitsch and Rössler 1989).
The plots above show evolution of the point for parameters (left) and (right).
The Hénon map has correlation exponent (Grassberger and Procaccia
1983) and capacity dimension (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
(3)
(4)
(Hénon 1969), which is one of the simplest two-dimensional invertible maps.
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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 D9, 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 D14, 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. & Graphics13,
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,
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Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 95-97, 1991.
and 
.
and 
colored according to the number of iterations required
to escape (Michelitsch and Rössler 1989).
for parameters 
(left) and 
(right).
(Grassberger and Procaccia
1983) and capacity dimension 
(Russell et al. 1980). Hitzl and Zele (1985)
give conditions for the existence of periods 1 to 6.
