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| [[Image:Arnoldcatmap.svg|thumb|Picture showing how the linear map stretches the unit square and how its pieces are rearranged when the [[modulo operation]] is performed. The lines with the arrows show the direction of the contracting and expanding [[eigenspace]]s]]
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| In [[mathematics]], '''Arnold's cat map''' is a [[chaos theory|chaotic]] map from the [[torus]] into itself, named after [[Vladimir Arnold]], who demonstrated its effects in the 1960s using an image of a [[cat]], hence the name.<ref name="Arnold"/>
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| Thinking of the torus <math>\mathbb{T}^2</math> as the [[Quotient space (linear algebra)|quotient space]] <math>\mathbb{R}^2/\mathbb{Z}^2</math> Arnold's cat map is the transformation <math>\Gamma : \mathbb{T}^2 \to \mathbb{T}^2</math> given by the formula
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| :<math>\Gamma \, : \, (x,y) \to (2x+y,x+y) \bmod 1.</math>
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| Equivalently, in [[matrix (mathematics)|matrix]] notation, this is
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| :<math>\Gamma \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \bmod 1 = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \bmod 1.</math>
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| That is, with a unit size equal to the width of the square image, the image is [[Shear mapping|sheared]] one unit up, then one unit to the right, and all that lies outside that unit square is shifted back by the unit until it's within the square.
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| ==Properties==
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| * Γ is [[invertible function|invertible]] because the matrix has [[determinant]] 1 and therefore its [[Integer_matrix#Properties|inverse has integer entries]],
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| * Γ is [[Measure-preserving dynamical system|area preserving]],
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| * Γ has a unique [[hyperbolic fixed point]] (the [[Vertex (geometry)|vertices]] of the square). The linear transformation which defines the map is hyperbolic: its [[eigenvalue]]s are irrational numbers, one greater and the other smaller than 1 (in absolute value), so they are associated respectively to an expanding and a contracting [[eigenspace]] which are also the [[stable manifold|stable and unstable manifolds]]. The eigenspace are orthogonal because the matrix is [[symmetric matrix|symmetric]]. Since the eigenvectors have [[rational dependence|rationally independent]] components both the eigenspaces [[dense set|densely]] cover the torus. Arnold's cat map is a particularly well-known example of a ''[[hyperbolic]] toral automorphism'', which is an [[automorphism]] of a [[torus]] given by a square [[unimodular matrix]] having no [[eigenvalues]] of absolute value 1.<ref name="Franks"/>
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| * The set of the points with a [[periodic orbit]] is [[dense set|dense]] on the torus. Actually a point is preperiodic if and only if its coordinates are [[rational number|rational]].
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| * Γ is [[Topological transitivity|topologically transitive]] (i.e. there is a point whose orbit is [[dense set|dense]], this happens for example for any points on the expanding [[eigenspace]])
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| * The number of points with period ''n'' is exactly |λ<sub>1</sub><sup>''n''</sup> + λ<sub>2</sub><sup>''n''</sup>−2| (where λ<sub>1</sub> and λ<sub>2</sub> are the eigenvalues of the matrix). For example, the first few terms of this series are 1, 5, 16, 45, 121, 320, 841, 2205 ....<ref>{{SloanesRef|sequencenumber=A004146}}</ref> (The same equation holds for any unimodular hyperbolic toral automorphism if the eigenvalues are replaced.)
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| * Γ is [[ergodic]] and [[Mixing (physics)|mixing]],
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| * Γ is an [[Anosov diffeomorphism]] and in particular it is [[Structural stability|structurally stable]].
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| == The discrete cat map ==
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| [[Image:Arnold cat.png|right|frame|From order to chaos and back.
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| Sample mapping on a picture of 150x150 pixels. The numbers shows the
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| iteration step. After 300 iterations arriving at the original image]]
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| [[File:Arnold's Cat Map animation (74px, zoomed, labelled).gif|right|frame|Sample mapping on a picture of a pair of cherries. The image is 74 pixels wide, and takes 114 iterations to be restored, although it appears upside-down at the halfway point.]]
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| It is possible to define a discrete analogue of the cat map. One of this map's features is that image being apparently randomized by the transformation but returning to its original state after a number of steps. As can be seen in the picture to the right, the original image of the cat is [[shear mapping|sheared]] and then wrapped around in the first iteration of the transformation. After some iterations, the resulting image appears rather [[randomness|random]] or disordered, yet after further iterations the image appears to have further order—ghost-like images of the cat, multiple smaller copies arranged in a repeating structure and even upside-down copies of the original image—and ultimately returns to the original image.
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| The discrete cat map describes the [[phase space]] flow corresponding to the discrete dynamics of a bead hopping from site ''q''<sub>t</sub> (0 ≤ ''q''<sub>t</sub> < N) to site ''q''<sub>t+1</sub> on a circular ring with circumference ''N'', according to the [[second order equation]]:
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| :<math>q_{t+1} - 3q_{t} + q_{t-1} = 0 \mod N</math> | |
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| Defining the momentum variable ''p''<sub>t</sub> = ''q''<sub>t</sub> - ''q''<sub>t-1</sub>, the above second order dynamics can be re-written as a mapping of the square 0 ≤ ''q'', ''p'' < ''N'' (the [[phase space]] of the discrete dynamical system) onto itself:
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| :<math>q_{t+1} = 2q_{t} + p_{t} \mod N</math>
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| :<math>p_{t+1} = q_{t} + p_{t} \mod N</math> | |
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| This Arnold cat mapping shows [[mixing (physics)|mixing]] behavior typical for chaotic systems. However, since the transformation has a [[determinant]] equal to unity, it is [[area-preserving maps|area-preserving]] and therefore [[invertible function|invertible]] the inverse transformation being:
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| :<math>q_{t-1} = q_{t} - p_{t} \mod N</math>
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| :<math>p_{t-1} = -q_{t} + 2p_{t} \mod N</math>
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| For real variables ''q'' and ''p'', it is common to set ''N'' = 1. In that case a mapping of the unit square with periodic boundary conditions onto itself results.
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| When N is set to an integer value, the position and momentum variables can be restricted to integers and the mapping becomes a mapping of a toroidial square grid of points onto itself. Such an integer cat map is commonly used to demonstrate [[mixing (physics)|mixing]] behavior with [[Poincaré recurrence theorem|Poincaré recurrence]] utilising digital images. The number of iterations needed to restore the image can be shown never to exceed 3N.<ref name="DysonFalk"/>
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| For an image, the relationship between iterations could be expressed as follows:
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| :<math>
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| \begin{array}{rrcl}
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| n=0: \quad & T^0 (x,y) &= & \mbox{Input Image}(x,y) \\
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| n=1: \quad & T^1 (x,y) &= & T^0 \left( \bmod(2x+y, N), \bmod(x+y, N) \right) \\
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| & &\vdots \\
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| n=k: \quad & T^k (x,y) &= & T^{k-1} \left( \bmod(2x+y, N), \bmod(x+y, N) \right) \\
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| & &\vdots \\
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| n=m: \quad & \mbox{Output Image}(x,y) &=& T^m (x,y)
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| \end{array}
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| </math>
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| ==See also==
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| {{Portal|Mathematics}}
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| * [[List of chaotic maps]]
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| * [[Recurrence plot]]
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| ==References==
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| <references>
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| <ref name="Arnold">{{Fr icon}} {{cite book
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| | author= [[Vladimir I. Arnold]]
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| | coauthors=A. Avez
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| | title=Problèmes Ergodiques de la Mécanique Classique
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| | location=Paris
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| | publisher=Gauthier-Villars
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| | year=1967}};
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| '''English translation:''' {{cite book
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| |author=V. I. Arnold
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| |coauthors=A. Avez
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| |title=Ergodic Problems in Classical Mechanics
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| |location=New York
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| |publisher=Benjamin
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| |year=1968
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| }}</ref>
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| <ref name="Franks">{{cite journal
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| | last1 = Franks
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| | first1 = John M
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| | title = Invariant sets of hyperbolic toral automorphisms
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| | journal = American Journal of Mathematics
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| | volume = 99
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| | issue = 5
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| |date=October 1977
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| | pages = 1089–1095
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| | doi = 10.2307/2374001
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| | issn = 0002-9327
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| | publisher = The Johns Hopkins University Press
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| }}</ref>
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| <ref name="DysonFalk">{{cite journal
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| | title = Period of a Discrete Cat Mapping
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| | first1 = Freeman John
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| | last1 = Dyson
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| | authorlink1 = Freeman Dyson
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| | first2 = Harold
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| | last2 = Falk
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| | journal = The American Mathematical Monthly
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| | volume = 99
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| | issue = 7
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| | year = 1992
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| | pages = 603–614
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| | issn = 0002-9890
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| | jstor = 2324989
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| | publisher = Mathematical Association of America
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| }}</ref>
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| </references>
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| ==External links==
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| *{{MathWorld|urlname=ArnoldsCatMap|title=Arnold's Cat Map}}
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| *[http://www.mpipks-dresden.mpg.de/mpi-doc/kantzgruppe/wiki/projects/Recurrence.html Effect of randomisation of initial conditions on recurrence time]
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| * [http://demonstrations.wolfram.com/ArnoldsCatMap/ Arnold's Cat Map] by Enrique Zeleny, [[The Wolfram Demonstrations Project]].
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| {{Chaos theory}}
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| {{DEFAULTSORT:Arnold's Cat Map}}
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| [[Category:Chaotic maps]]
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| [[Category:Exactly solvable models]]
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Wilber Berryhill is what his wife enjoys to call him and he totally enjoys this name. To perform domino is some thing I really appreciate performing. North Carolina is the location he loves most but now he is considering other choices. Office supervising is my occupation.
Check out my website :: psychic readings online - http://203.250.78.160/,