Periods of the discretized Arnold Cat Map and its extension to n dimensions

Details Periods of the discretized Arnold Cat Map and its extension to n dimensions Periods of the discretized Arnold Cat Map and its extension to n dimensions

2011-11-13

arxiv.org/abs/1111.2984v1

Mathematics

Dynamical Systems

Scientific paper

A discrete dynamical system known as Arnold's Discrete Cat Map (Arnold's DCM) is given by (x_t+1, y_t+1) = (x_t + y_t, x_t + 2y_t) mod N; which acts on a two-dimensional square coordinate grid of size Nx?N. The de?ning characteristic of this map is that it has the property that when the NxN grid is a picture whose pixels are assigned (x,y) coordinates, the map scrambles the picture with each iteration. After a ?finite number of iterations, the picture is restored to its original shape and order. The number of iterations M, needed to restore the image, has a mysterious dependence on N. This period, as we will fi?nd out, is directly related to the divisibility of the Fibonacci numbers. We will exploit this property to show that, for any N, an image is not dense in itself. In the second half of the paper, we build on the work of Chen, Mao, and Chui to extend the DCM to three dimensions. Finally, we de?fine the generalized n dimensional DCM by introducing the idea of a "matrix union".

Affiliated with

Also associated with

No associations

Rating

Periods of the discretized Arnold Cat Map and its extension to n dimensions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Periods of the discretized Arnold Cat Map and its extension to n dimensions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Periods of the discretized Arnold Cat Map and its extension to n dimensions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-340353