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MathGroup Archive 2001

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chaos-to -order transform

  • To: mathgroup at smc.vnet.net
  • Subject: [mg27308] chaos-to -order transform
  • From: Roberto Brambilla <rlbrambilla at cesi.it>
  • Date: Tue, 20 Feb 2001 03:05:17 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

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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