Re: defining function on mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg22293] Re: defining function on mathematica
• From: Brian Higgins <bghiggins at ucdavis.edu>
• Date: Wed, 23 Feb 2000 01:01:25 -0500 (EST)
• References: <88snik\$ipe@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <88snik\$ipe at smc.vnet.net>,
Elisha Kobre <ekobre at erols.com> wrote:
> Hi,
> problem:
>
> I have a family of functions indexed by w where w lies in the interval
> [0,1). The functions are f[x_]:= x+ w +1/1000 Sin[2 Pi x]. I want to
> define (AND PLOT) a function h[w] where for each value of w, h[w] is
the
> 1000th iterate of f (with the specific w inserted into f) on x=0
> ,divided by 1000 ---(this is supposed to approximate the so called
> rotation number). The graph of h[w] is called a devils staircase. How
> can we define h so it works and we get a good plot ?
>
> I have tried nest, fold etc.... but run into problems with all of
them.
>
> I'll Be glad to provide more details.
>
> Thanks
>
> Elisha Kobre
>
Elisha,
The following code appears to produce the Devil's staircase from the
rotation map:
f[x_, w_] := x + w + (0.2) Sin[2 Pi x]
h[w_] := Nest[f[#, w] &, .1, 1000]
data = Table[{w, h[w]}, {w, 0, 1, .005}];
ListPlot[data, PlotJoined -> True, PlotRange -> All]

You can also use Plot but is not efficient (takes a long time to plot)

Plot[h[w], {w, 0, 1}, PlotPoints -> 30]

Note that if the factor in front of the Sin function is reduced to
1/1000 as you suggest, then the structure of the staircase is hardly
noticeable.

Cheers,
Br

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