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MathGroup Archive 1996

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Map Attractors in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg3041] Map Attractors in Mathematica
  • From: Jonathan Lee <bobjon at gatewest.net>
  • Date: Thu, 25 Jan 1996 03:21:08 -0500
  • Organization: Gate West Communications, Winnipeg, Manitoba, Canada

I am looking for a method to display chaotic map attractors in Mathematica. The idea 
is to take some rectangular area (usually a unit square) with some shape in it and 
continually deform the rectangle. Some examples are Henon's attractor, the Baker's 
transformation, and Arnol'd's cat map. In each case, each point (x_n, y_n) is moved 
to a new point by the relation (x_n+1, y_n+1)=(f(x_n, y_n), g(x_n, y_n)). For 
example with the Arnol'd's cat map the transformation goes along these lines: 
1)start with a unit square with some shape in it (traditionally a cat's face) then 
2)stretch the unit square to twice it's original width and three times it's original 
height, finally 3)cut up the 2*3 rectangle into 6 unit squares and mash them 
together (layer them one on top of another). This process is continually done to 
produce more complex maps. In short the transformation is this

x_n+1=x_n+y_n mod 1
y_n+1=x_n+2 y_n mod 1

Does anyone have any suggestions of how I can apply this transformation to each 
point in the square. Any help is appreciated. Thanks in advance.

--Jonathan Lee


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