The name Hénon map refers to at least two maps .
The first is the two-dimensional dissipative quadratic
map given by the coupled equations
(Hénon 1976).
The strange attractor illustrated above is obtained for
Writing the first map as Jacobian
matrix and Jacobian determinant are
It is therefore invertible when
(5)
The map scales signed area by area by iteration (Liang
2016). Consequently, whenever its two Lyapunov
characteristic exponents exist, their sum is given by
(6)
(Eckmann and Ruelle 1985). In particular, the standard value map area-contracting.
The illustration above shows two regions of space for the map with iterations
required to escape (Michelitsch and Rössler 1989).
The plots above show evolution of the point parameters
The Hénon map has correlation exponent capacity dimension et al. 1980). Hitzl and Zele (1985)
give conditions for the existence of periods 1 to 6.
Trott (2006) uses the normalization
(7)
For coordinate
transformation map
to this form with real fixed points are
(8)
A real fixed point therefore exists iff
A point least period 3 under iff
(9)
where iteration .
The equation fixed points . Eliminating the coordinates of a real point quantifier elimination gives conditions
on the parameters
(10)
whose three distinct points have least period 3. Endler and Gallas (2002) give a polynomial
of polynomial degree 4 that parametrizes orbits
of least period 4.
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 ,
Signed Area
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References Dickau, R. M. "The Hénon Attractor." https://www.robertdickau.com/henon.html . Eckmann,
J.-P. and Ruelle, D. "Ergodic Theory of Chaos and Strange Attractors."
Rev. Mod. Phys. 57 , 617-656, 1985. https://doi.org/10.1103/RevModPhys.57.617 . Endler,
A. and Gallas, J. A. C. "Arithmetical Signatures of the Dynamics of
the Hénon Map." Phys. Rev. E 65 , 036231, 2002. https://doi.org/10.1103/PhysRevE.65.036231 . Gleick,
J. Chaos:
Making a New Science. 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. Liang, X. S. "Information Flow
and Causality as Rigorous Notions Ab Initio." Phys. Rev. E 94 ,
052201, 2016. https://doi.org/10.1103/PhysRevE.94.052201 . 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 Morosawa, S.; Nishimura, Y.; Taniguchi, M.; and Ueda, T. "Dynamics
of Generalized Hénon Maps." Ch. 7 in Holomorphic
Dynamics. Peitgen, H.-O. and Richter, D. H. The
Beauty of Fractals: Images of Complex Dynamical Systems. Peitgen, H.-O. and Saupe, D. (Eds.). "A Chaotic Set in the
Plane." §3.2.2 in The
Science of Fractal Images. Russell,
D. A.; Hanson, J. D.; and Ott, E. "Dimension of Strange Attractors."
Phys. Rev. Let. 45 , 1175-1178, 1980. Trott, M. "Polynomials
in Inequalities." §1.2.3 in The
Mathematica GuideBook for Symbolics. https://doi.org/10.1007/0-387-28815-5 .
https://www.mathematicaguidebooks.org/ . Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Hénon Map
Cite this as:
Weisstein, Eric W. "Hénon Map."
From MathWorld https://mathworld.wolfram.com/HenonMap.html
Subject classifications
and 
.
, its Jacobian
matrix and Jacobian determinant are
![[-2alphax 1; beta 0]](/images/equations/HenonMap/Inline12.svg)
, with
and area by 
at every iteration (Liang
2016). Consequently, whenever its two Lyapunov
characteristic exponents exist, their sum is given by
makes the map area-contracting.
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.
, the linear coordinate
transformation 
,
converts the first map
to this form with 
and 
. For real 
and 
, its fixed points are 
, where
.
has least period 3 under 
iff
denotes the third iteration.
The equation 
alone also admits fixed points. Eliminating the coordinates of a real point 
by quantifier elimination gives conditions
on the parameters 
and 
.
For example, 
and 
give the orbit
