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>
• 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
>>
>>alternative email: rlbtftn at netscape.net
>>
>>
>
>
>

--
Respectfully, Roger L. Bagula