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