Mean difference

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 eigenspaces

In mathematics, Arnold's cat map is a 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.

Thinking of the torus $\mathbb {T} ^{2}$ as the quotient space $\mathbb {R} ^{2}/\mathbb {Z} ^{2}$ Arnold's cat map is the transformation $\Gamma :\mathbb {T} ^{2}\to \mathbb {T} ^{2}$ given by the formula

$\Gamma \,:\,(x,y)\to (2x+y,x+y){\bmod {1}}.$

Equivalently, in matrix notation, this is

$\Gamma \left({\begin{bmatrix}x\\y\end{bmatrix}}\right)={\begin{bmatrix}2&1\\1&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}{\bmod {1}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&0\\1&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}{\bmod {1}}.$

That is, with a unit size equal to the width of the square image, the image is 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.

Properties

• Γ is invertible because the matrix has determinant 1 and therefore its inverse has integer entries,
• Γ is area preserving,
• Γ has a unique hyperbolic fixed point (the vertices of the square). The linear transformation which defines the map is hyperbolic: its eigenvalues 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 and unstable manifolds. The eigenspace are orthogonal because the matrix is symmetric. Since the eigenvectors have rationally independent components both the eigenspaces 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.
• The set of the points with a periodic orbit is dense on the torus. Actually a point is preperiodic if and only if its coordinates are rational.
• Γ is topologically transitive (i.e. there is a point whose orbit is dense, this happens for example for any points on the expanding eigenspace)
• The number of points with period n is exactly |λ₁ⁿ + λ₂ⁿ−2| (where λ₁ and λ₂ are the eigenvalues of the matrix). For example, the first few terms of this series are 1, 5, 16, 45, 121, 320, 841, 2205 .... (The same equation holds for any unimodular hyperbolic toral automorphism if the eigenvalues are replaced.)
• Γ is ergodic and mixing,
• Γ is an Anosov diffeomorphism and in particular it is structurally stable.

The discrete cat map

From order to chaos and back. Sample mapping on a picture of 150x150 pixels. The numbers shows the iteration step. After 300 iterations arriving at the original image

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.

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 sheared and then wrapped around in the first iteration of the transformation. After some iterations, the resulting image appears rather 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.

The discrete cat map describes the phase space flow corresponding to the discrete dynamics of a bead hopping from site q_t (0 ≤ q_t < N) to site q_t+1 on a circular ring with circumference N, according to the second order equation:

$q_{t+1}-3q_{t}+q_{t-1}=0\mod N$

Defining the momentum variable p_t = q_t - q_t-1, 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:

$q_{t+1}=2q_{t}+p_{t}\mod N$

$p_{t+1}=q_{t}+p_{t}\mod N$

This Arnold cat mapping shows mixing behavior typical for chaotic systems. However, since the transformation has a determinant equal to unity, it is area-preserving and therefore invertible the inverse transformation being:

$q_{t-1}=q_{t}-p_{t}\mod N$

$p_{t-1}=-q_{t}+2p_{t}\mod N$

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.

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 behavior with Poincaré recurrence utilising digital images. The number of iterations needed to restore the image can be shown never to exceed 3N.

For an image, the relationship between iterations could be expressed as follows:

${\begin{array}{rrcl}n=0:\quad &T^{0}(x,y)&=&{\mbox{Input Image}}(x,y)\\n=1:\quad &T^{1}(x,y)&=&T^{0}\left({\bmod {(}}2x+y,N),{\bmod {(}}x+y,N)\right)\\&&\vdots \\n=k:\quad &T^{k}(x,y)&=&T^{k-1}\left({\bmod {(}}2x+y,N),{\bmod {(}}x+y,N)\right)\\&&\vdots \\n=m:\quad &{\mbox{Output Image}}(x,y)&=&T^{m}(x,y)\end{array}}$

See also

Sportspersons Hyslop from Nicolet, usually spends time with pastimes for example martial arts, property developers condominium in singapore singapore and hot rods. Maintains a trip site and has lots to write about after touring Gulf of Porto: Calanche of Piana.

• List of chaotic maps
• Recurrence plot

External links

• 
  
  I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
  
  
  
  Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.
• Effect of randomisation of initial conditions on recurrence time
• Arnold's Cat Map by Enrique Zeleny, The Wolfram Demonstrations Project.