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MathGroup Archive 2001

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Re: chaos-to -order transform

  • To: mathgroup at smc.vnet.net
  • Subject: [mg27321] Re: chaos-to -order transform
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Wed, 21 Feb 2001 03:16:50 -0500 (EST)
  • Organization: Universitaet Leipzig
  • References: <96t8ss$pqi@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,

a) revrite you code to

T[t_] := {{2 Cos[t], Cos[t] - Sin[t]}, {Cos[t] + Sin[t], Cos[t]}}

Iterate[p_, t_?NumericQ, n_Integer] := With[{tau = T[t]},
    NestList[Mod[tau.#, 1] &, p, n]
    ]


p = {.4, .7};
t = 1.14724;(*parameter 0 <= t <= Pi/2*)
lst = Iterate[p,t, 500];

ListPlot[lst, Frame -> True, Axes -> False, AspectRatio -> 1]

b) Select the plot, hold down the Mod1 (Ctrl on Windows)
   key, click the coordinate with the mouse, type
   Mod1-C  (the Mod1 key is already pressed) move
   to the input line you like and paste the coordinates
   with Mod1-V

Regards
  Jens



Roberto Brambilla wrote:
> 
> Dear math-friends,
> some years ago, studying transitions from chaos to order,
> I and a my collegue (Casartelli), devised the following example:
>  (1) consider the positive unit square Q{x,y|0<=(x,y)<1}
>  (2) consider the following real unitary transform of Q into itself
> 
>   T[t_]={{2 Cos[t],Cos[t]-Sin[t]},
>          {Cos[t]+Sin[t],Cos[t]}}
> 
>      {x1,y1} = T.{x0,y0} (*mod 1*)
> 
>       i.e. take only the decimal part.
> 
> Then, given a value of parameter t (0<t<Pi/2) and a starting
> arbitrary point (seed) {x0,y0} in Q, build a long list (say some
> hundreds)of points
>   {{x0,y0},{x1,y1},.....{xn,yn}}
> where each point is the transformed of the preceding one.
> Plotting these lists you obtain very different images depending
> on the value of the parameter t and the starting points.
> It is useful to put in a single graph many plots corresponding to the
> same t value an different starting points (fine with different colors!).
> For small value of t, points seem to be scattered in Q in a random
> o quasi-random way (chaos).
> For t near Pi/2 points arrange in circles (order).
> For intermediate t values you will obtain complex figures.
> Little changes in t sometimes can produce completely different
> figures.
> (3) Problem : for which t values significant transitions happen?
> 
>                          -o0o-
> 
> At that time I used a C-program and for a fixed t and list length,
> I could choose the starting point with the mouse, directly clicking
> on a unit square picture and I could follow in real time the outspreading
> of points into the square.
> I had no to memorize the sequences, since points were represented
> (memorized) on the screen. All worked very quicly.
> 
> Using Mathematica is possible something analogue?
> Actually I use the rfollowing rather unsatisfactory method,
> since slow and memory consuming and no movie-effect :
> 
> t=1.14724; (*parameter 0<=t<=Pi/2*)
> T={{2 Cos[t],Cos[t]-Sin[t]},{Cos[t]+Sin[t],Cos[t]}};
> 
> lmax=500;
> p={.4,.7}; (*seed,initial point*)
> l={p};
> For[k=1,k<=lmax,k++,
>     p=Mod[T.p,1];
>     AppendTo[l,p];
> ]
> ListPlot[l,Frame->True,Axes->False,AspectRatio->1]
> 
> Then with Show[] I put figures together etc..
> Is it possible to choose the seed on the figure and
> imitate in some way the old C program?
> 
> Bye, 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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