Re: the circle map

*To*: mathgroup at smc.vnet.net*Subject*: [mg52287] Re: the circle map*From*: Roger Bagula <tftn at earthlink.net>*Date*: Sun, 21 Nov 2004 07:23:22 -0500 (EST)*References*: <cneuml$qbc$1@smc.vnet.net> <cnn0rm$8pf$1@smc.vnet.net>*Reply-to*: tftn at earthlink.net*Sender*: owner-wri-mathgroup at wolfram.com

Dear Peter Valko, As I explained map is "sensative" to both the oringinal a0 ( angle) and the initial conditions. It is easy to get the dengerate circle, and not so easy to get some of the pretty effects This also a problem with the Martin map: I got in real trouble in a Martin map egroup I was in because I was trying to get the simple degenete sets, ha, ha. On the fractional part: I've always got : Mod[x.1]=x-Floor[x] to give the same output. If it doesn't, it may be one more problem with the Mod[x,1] algorithm in Mathematica ( to get Modulo ones where continued use of Sqrt[] are involved , you have to use N[ Mod[x,1], digits_accuracy]. It shouldn't be a problem where a number not a function is used like you have been doing. Mathematica has a bad habit of carrying what errors it does make on and building on them if "exact" numbers as symbols in a list are used. Peter Valko wrote: >Roger, >Can you tell me why is it that in the following code of yours: > >Clear[x, y, n]; >a0 = 0.41209; >x[n_] := x[n] = Mod[-a0*x[n - 1] - y[n - 1], 1]; >y[n_] := y[n] = Mod[x[n - 1], 1] ; >x[0] = 0.7; >y[0] = .65; >a = Table[{x[n], y[n]}, {n, 0, 10000}]; >ListPlot[a, PlotRange -> All] ; > >we get a fractal-like pic, but changing to a0 = 0.41208 we do not? >(And why is that replacing Mod[-,1] by FractionalPart[-] in the above >code will not give the same phenomenon? > >Peter > > >Roger Bagula <tftn at earthlink.net> wrote in message news:<cneuml$qbc$1 at smc.vnet.net>... > > >>I've done a lot of searches on chaos >> and Mathematica and have never seem this. >>It is sensative chaos , in both the angle based a0 and the >>initial starting point. >>The circle was used by Chua as a starting point in his lectures on Chaos. >> >>Clear[x,y,a,b,s,g,a0] >>(* circle map: from Chaos in Digital Filters ,Chua,Lin, >> IEEE transactions on Circuits and Systems,vol 35 no 6 June 1988*) >> (* very sensitive to intial conditions*) >>a0=Cos[Pi/6]/2; >>x[n_]:=x[n]=Mod[-a0*x[n-1]-y[n-1],1] >>y[n_]:=y[n]=Mod[x[n-1],1] >>x[0]=0.7;y[0]=.65; >>a=Table[{x[n],y[n]},{n,0, 10000}]; >>ListPlot[a, PlotRange->All] >> >>Respectfully, Roger L. Bagula >> >>tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : >>alternative email: rlbtftn at netscape.net >>URL : http://home.earthlink.net/~tftn >> >> > > > -- Respectfully, Roger L. Bagula tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn at netscape.net URL : http://home.earthlink.net/~tftn