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MathGroup Archive 2001

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Chaos-to-order (cont'd)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg27367] Chaos-to-order (cont'd)
  • From: Roberto Brambilla <rlbrambilla at cesi.it>
  • Date: Thu, 22 Feb 2001 02:25:11 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,
refeerring to my previous e-mail [mg27308]
and taking into account suggestions of some of 
you, I have the points orbit in te unit square 

T[t_]:={{2 Cos[t],Cos[t]-Sin[t]},{Cos[t]+Sin[t],Cos[t]}};
(*0<=t<=pi/2*)
Orbit[t_,p_,n_Integer] := With[{tau = T[t]},
    NestList[Mod[tau.#, 1] &, p, n]
    ]


es. two orbits generate by two different initial points:

t0=Pi/3.;
l1=Orbit[t0,{.4,.7},1000];
 ListPlot[l1,Frame->True,Axes->False,AspectRatio->1,
 PlotStyle->Hue[.9]]

l2=Orbit[t0,{.5,.2},1000];
 ListPlot[l1,Frame->True,Axes->False,AspectRatio->1,
 PlotStyle->Hue[.7]]

The two figures looks very different.
Both lists contain 1001 points, but the first suggests that there are
returning points, i.e a finite sequence, or closed orbit. 
In a more coarse precision the first list would have repeated points.
I think that a quick way to detect closed orbits
(or quasi closed orbits) may be counting colored pixels
in the unit square and counting the background white pixels.
But I have not found this kind of functions.
Any idea?
Many thanks, Roberto 
Roberto Brambilla
CESI
Via Rubattino 54
20134 Milano
tel +39.2.2125.5875
fax +39.2.2125.610
rlbrambilla at cesi.it



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