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Re:chaos-to -order transform

• To: mathgroup at smc.vnet.net
• Subject: [mg27379] Re:chaos-to -order transform
• From: bghiggins at ucdavis.edu (Brian Higgins)
• Date: Thu, 22 Feb 2001 02:25:25 -0500 (EST)
• Organization: The Math Forum
• References: <96t8ss$pqi@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Roberto, I was playing around with your function some more and came to
realize why it was desirable to see how different initial points
evolve doing the iteratio. So here is another function that is  color
coded and all plots are displayed within the same frame. The initial
points are generated at random:

First load these packages

In[1]:=<< Graphics`MultipleListPlot`

In[2]:=<< Graphics`Colors`

Then evaluate the module

Newiterator[npts_, t_] := 
  Module[{init, colorpts, InitPts, T, datalists}, 
    init = Table[{Random[], Random[]}, {npts}];
    colorpts = 
      Map[ToExpression[AllColors[[#]]] &, 
        Table[Random[Integer, {1, 193}], {npts}]];
    InitPts = Transpose[{colorpts, Map[Point[#] &, init]}];
    T = {{2 Cos[t], Cos[t] - Sin[t]},
         {Cos[t] + Sin[t], Cos[t]}} ;
    datalists = 
      Map[NestList[(Mod[T.{#[[1]], #[[2]]}, 1]) &, #, 200] &, init];
    MultipleListPlot[datalists, SymbolStyle -> colorpts, 
      SymbolShape -> MakeSymbol[RegularPolygon[20, 1]], Frame -> True,

      Axes -> False, AspectRatio -> 1, Epilog -> {PointSize[0.03],
InitPts}]]

Newiterator[6, 1.56]

Please note if you attempt to plot 4 or more points you may need to
increase the size of memory for the kernel.

If you have a reference for any analysis on your map, I would
appreciate getting it. It is a lovely example for showing students
studying nonlinear dynamics  the various attraction basins for a 
finite dimensional map.

Cheers,

Brian