> 
# mapiter.ms  
# Graphical iteration of 1-d maps
# 
> mapiter := proc(F, x0, a, b, skipiter, plotiter, caption)
>    local xp, k, p, orbit, curve, diag;
> 
>    xp := x0; # starting value
> 
>    # Skip the first few iterations if long term behaviour is desired.
>    # To plot all iterations use 0 for skipiter.
> 
>    p := array(0..plotiter);
>    for k from 1 to skipiter do xp := F(xp); od:
>    for k from 0 to plotiter do p[k] := xp; xp := F(xp); od;
> 
>    curve := plot(F(x), x=a..b, color=blue);
>    diag := plot(x, x=a..b, color=RGB(0,0,0));
>    orbit := plot([seq(op([[p[k-1],p[k]],[p[k],p[k]]]),
> k=1..plotiter)],
>             color=red);
>    plots[display]([orbit, curve,
> diag],title=caption,scaling=CONSTRAINED);
> end:


# Test the procedure on the logistic map
> Q := (x) -> a*x*(1-x);

                        Q := x -> a x (1 - x)

> # a := 2.9:  mapiter(Q, 0.1, 0, 1, 0, 100,`a=2.9`);    # period 1  
> # a := 3.2:  mapiter(Q, 0.1, 0, 1, 0, 100,`a=3.2`);    # period 2  
> # a := 3.5:  mapiter(Q, 0.1, 0, 1, 1000, 100,`a=3.5`); # period 4  
> # a := 3.56: mapiter(Q, 0.1, 0, 1, 1000, 100,`a=3.56`); # period 8  
> # a := 3.74: mapiter(Q, 0.1, 0, 1, 1000, 100,`a=3.74`); # period 5
> #interface(plotdevice=postscript,plotoutput=`mapiter1.ps`);
> a := 3.84: mapiter(Q, 0.1, 0, 1, 1000, 100,`a=3.84`); # period 3


> #interface(plotdevice=postscript,plotoutput=`mapiter2.ps`);
> a := 3.9:  mapiter(Q, 0.1, 0, 1, 1000, 100,`a=3.9`); # chaos


# Try some other maps
> # mapiter(cos, 0.1, 0, 1, 0, 100,`cos(x)`); 
> # mapiter((x) -> exp(-x), 0.1, 0, 1, 0, 100,`exp(-x)`); 
> # mapiter((x) -> arctan(x), 0.5, -1, 1, 0, 100,`arctan(x)`);