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MathGroup Archive 2004

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Re: the circle map

  • To: mathgroup at smc.vnet.net
  • Subject: [mg52292] Re: the circle map
  • From: Peter Pein <petsie at arcor.de>
  • Date: Sun, 21 Nov 2004 07:23:28 -0500 (EST)
  • References: <cneuml$qbc$1@smc.vnet.net> <cnn0rm$8pf$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Peter Valko wrote:
> Roger,
> Can you tell me why is it that in the following code of yours: 
> 
> Clear[x, y, n];
> a0 = 0.41209;
> x[n_] := x[n] = Mod[-a0*x[n - 1] - y[n - 1], 1];
> y[n_] := y[n] = Mod[x[n - 1], 1] ;
> x[0] = 0.7;
> y[0] = .65;
> a = Table[{x[n], y[n]}, {n, 0, 10000}]; 
> ListPlot[a, PlotRange -> All] ;
> 
> we get a fractal-like pic, but changing to a0 = 0.41208 we do not?

This behaviour is typical for chaotic systems. See for instance 
http://mathworld.wolfram.com/LogisticMap.html

> (And why is that replacing Mod[-,1] by FractionalPart[-] in the above
> code will not give the same phenomenon?

(#1[-a0*0.7 - 0.65] & ) /@  { Mod[#1, 1] & , FractionalPart }

{0.06153699999999995, -0.938463}

> 
> Peter
> 
> 


-- 
Peter Pein
10245 Berlin


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