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Elliptic curve point multiplication

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[[Image:HenonMapImage.png|thumb|Hénon attractor for ''a'' = 1.4 and ''b'' = 0.3]]
[[Image:Henon Multifractal Map movie.gif|thumb|Hénon attractor for ''a'' = 1.4 and ''b'' = 0.3]]
The '''Hénon map''' is a discrete-time [[dynamical system]]. It is one of the most studied examples of dynamical systems that exhibit [[chaos theory|chaotic behavior]]. The Hénon map takes a point (''xn'', ''yn'') in the plane and maps it to a new point

:<math>\begin{align}
x_{n+1} &= y_n+1-a x_n^2,\\
y_{n+1} &= b x_n.
\end{align}</math>

The map depends on two parameters, ''a'' and ''b'', which for the '''classical Hénon map''' have values of ''a'' = 1.4 and ''b'' = 0.3. For the classical 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 [[orbit diagram]].

The map was introduced by [[Michel Hénon]] as a simplified model of the [[Poincaré map|Poincaré section]] of the [[Lorenz attractor|Lorenz model]]. For the classical map, an initial point of the plane will either approach a set of points known as the Hénon [[Attractor#Strange attractor|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 1.25 ± 0.02<ref name="Grassberger 1983">{{cite journal | author=P. Grassberger and I. Procaccia | title=Measuring the strangeness of strange attractors | journal=Physica | year=1983 | volume=9D | issue=1-2| pages=189–208| bibcode=1983PhyD....9..189G | doi=10.1016/0167-2789(83)90298-1 }}</ref> and a [[Hausdorff dimension]] of 1.261 ± 0.003<ref name="Russel 1980">{{cite journal | author=D.A. Russel, J.D. Hanson, and E. Ott | title=Dimension of strange attractors | journal=Physical Review Letters | year=1980 | volume=45 | issue=14 | pages=1175| doi= 10.1103/PhysRevLett.45.1175 | bibcode=1980PhRvL..45.1175R}}</ref> for the attractor of the classical map.

==Attractor==
[[Image:Henon bifurcation map b=0.3.png|thumb|right|Orbit diagram for the Hénon map with ''b=0.3''. Higher density (darker) indicates increased probability of the variable ''x'' acquiring that value for the given value of ''a''. Notice the ''satellite'' regions of chaos and periodicity around ''a=1.075'' -- these can arise depending upon initial conditions for ''x'' and ''y''.]]
The Hénon map maps two points into themselves: these are the invariant points. For the classical values of ''a'' and ''b'' of the Hénon map, one of these points is on the attractor:
: <math>x = \frac{\sqrt{609}-7}{28} \approx 0.631354477\dots\qquad y = \frac{3\left(\sqrt{609}-7\right)}{280} \approx 0.189406343\dots\,</math>
This point is unstable. Points close to this fixed point and along the slope 1.924 will approach the fixed point and points along the slope -0.156 will move away from the fixed point. These slopes arise from the linearizations of the [[stable manifold]] and [[unstable manifold]] of the fixed point. The unstable manifold of the fixed point in the attractor is contained in the [[strange attractor]] of the Hénon map.

The Hénon map does not have a strange attractor for all values of the parameters ''a'' and ''b''. For example, by keeping ''b'' fixed at 0.3 the bifurcation diagram shows that for ''a'' = 1.25 the Hénon map has a stable periodic orbit as an attractor.

Cvitanović et al. have shown how the structure of the Hénon strange attractor can be understood in terms of unstable periodic orbits within the attractor.

==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>.

==See also==
* [[Horseshoe map]]
* [[Takens' theorem]]

==Notes==
{{reflist}}

==References==
* {{cite journal | author=M. Hénon | title=A two-dimensional mapping with a strange attractor | journal=Communications in Mathematical Physics | year=1976 | volume=50 | issue=1 | pages=69–77 | doi=10.1007/BF01608556|bibcode = 1976CMaPh..50...69H }}
* {{cite journal
|journal = Physical Review A
|year = 1988
|title = Topological and metric properties of Hénon-type strange attractors
|pages = 1503–1520
|issue = 3
|author = Predrag Cvitanović, Gemunu Gunaratne, and Itamar Procaccia
|volume = 38
|doi = 10.1103/PhysRevA.38.1503
|pmid=9900529
|bibcode = 1988PhRvA..38.1503C }}
* {{cite journal | author=M. Michelitsch and O. E. Rössler | title=A New Feature in Hénon's Map | journal = Computers & Graphics | year=1989 | volume=13 | issue=2 | pages=263–265| url=http://mathworld.wolfram.com/HenonMap.html | doi=10.1016/0097-8493(89)90070-8}}. 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

==External links==
*[http://ibiblio.org/e-notes/Chaos/henon.htm Interactive Henon map] and [http://ibiblio.org/e-notes/Chaos/strange.htm Henon attractor] in [http://ibiblio.org/e-notes/Chaos/contents.htm Chaotic Maps]
* [http://complexity.xozzox.de/nonlinmappings.html Another interactive iteration of the Henon Map] by A. Luhn
* [http://demonstrations.wolfram.com/OrbitDiagramOfTheHenonMap// Orbit Diagram of the Hénon Map] by C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr, [[The Wolfram Demonstrations Project]].
* [http://memosisland.blogspot.de/2013/02/multicore-run-in-matlab-via-python.html Matlab code for the Hénon Map] by M.Suzen
* [http://experiences.math.cnrs.fr/L-attracteur-de-Henon.html Simulation] of [[Hénon map]] in javascript (experiences.math.cnrs.fr) by Marc Monticelli.

{{Chaos theory}}

{{DEFAULTSORT:Henon Map}}
[[Category:Chaotic maps]]