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chaos-to -order transform

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

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]},

     {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
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
(3) Problem : for which t values significant transitions happen?


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]}};

p={.4,.7}; (*seed,initial point*)

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
Via Rubattino 54
20134 Milano
tel +39.2.2125.5875
fax +39.2.2125.610
rlbrambilla at

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