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