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Re: Map Attractors in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg3056] Re: [mg3048] Map Attractors in Mathematica
*From*: Preston Nichols <nichols at godel.math.cmu.edu>
*Date*: Tue, 30 Jan 1996 02:21:38 -0500
*Sender*: owner-wri-mathgroup at wri.com
Jonathan Lee <bobjon at gatewest.net> in [mg3048] wrote:
>>>>
I am looking for a way to generate chaotic maps in Mathematica. The
idea is to plot the orbits of all points in a rectangular region
under a given transformation (x,y)->(f(x,y),g(x,y)). For example,
the Arnol'd's cat map based on the transformation (x,y)->(x+y mod
1,x+2y mod 1) is based on stretching out a unit square to twice it's
original width and three times it's original height to produce 6
unit squares which are all mashed together (basically they are
layered on top of one another). The trick is that the unit square
has some sort of shape in it (traditionally a cat's face), which is
also appropriately distorted.If anyone has any suggestions on how to
generate various iterations of this or any other map all and any
help is appreciated. Thanks in advance.
<<<<
I can offer the beginning of an answer.
A few months ago, I decided to see if I could invent an iterated
function system to reproduce some pictures in a paperback edition of
Jurrassic Park by Michel Crichton.
Here's the code I came up with to solve the exercise I had set
myself. Modifying it to iterate other functions may do what you
want. The complete notebook is available from:
http://www.contrib.andrew.cmu.edu/pdn/mmantbks.html
-----------------------------------------------
PointListQ[X_] := MatchQ[Dimensions[X], {_,2}];
onestep[X_?PointListQ] := Module[{y1=N[X],y2,m},
m = {{0,1},{-1,0}};
y1 = (# - y1[[1]])& /@ y1;
y2 = (m.#)& /@ y1;
Return[Reverse[y1]~Join~y2]
]
psteps[X_?PointListQ,1] :=
psteps[X,1] = onestep[X]
psteps[X_?PointListQ,p_?IntegerQ] :=
psteps[X,p] = onestep[psteps[X,p-1]]
ShowLines[X_?PointListQ] := Show[Graphics[Line[X]],
AspectRatio->Automatic]
(*
The function onestep, which is being iterated, takes a list of
points and appends to it a copy of itself rotated about its first
point. (The matrix m is the rotation matrix, and could easily be
made an input variable.)
This is designed for an initial set something like
start = {{0,0},{1,0}};
and then try, for example,
ShowLines[psteps[start,10]]
*)
-----------------------------------------------------
One could use
psteps[X_?PointListQ,p_?IntegerQ] := Nest[onestep, X, p],
which is logically equivalent to my code, but my way is faster if
you plan to generate pictures corresponding to various numbers of
iterations. This is because the assignment inside my definition of
psteps makes Mathematica "remember" previous results, which can then
be used "directly" when you ask for a larger number of iterations.
Unfortunately, this approach might backfire if you try to run
really large numbers of iterations, because Mathematica will have to
remember so much. (On the other hand, I don't really know whether
using Nest avoids that problem.)
Other than this, I haven't done anything at all that a more
sophisticated programmer might have done to make my code efficient.
-----------------------------------------------------
Good Luck!
Preston Nichols
Visiting Assistant Professor
Department of Mathematics
Carnegie Mellon University
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