       Re: the circle map

• To: mathgroup at smc.vnet.net
• Subject: [mg52267] Re: the circle map
• From: p-valko at tamu.edu (Peter Valko)
• Date: Sat, 20 Nov 2004 03:41:36 -0500 (EST)
• References: <cneuml\$qbc\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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 = 0.7;
y = .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=0.7;y=.65;
> a=Table[{x[n],y[n]},{n,0, 10000}];
> ListPlot[a, PlotRange->All]
>
> Respectfully, Roger L. Bagula
>