English: Critical orbits for f(z)=z^4-z.[1] Because critical orbit tends to fixed point very slowly there is a gap between critical orbit and fixed point.
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kill(all);
remvalue(all);
/*------------- functions definitions ---------*/
/* function */
f(z):=z^4 -z;
GiveListOfCriticalPoints(fun):=
block(
[d,s],
/* derivative */
d:diff(fun,z,1),
/* critical points z: d=0 */
s:solve(d=0,z),
/* remove "z=" from list s */
s:map('rhs,s),
/* convert to form x+y*%i */
s:map('rectform,s),
s:map('float,s),
return(s)
)$
/* f(z) is used as a global function
I do not know how to put it as a argument */
GiveOrbit(z0,OrbitLength):=
block(
[z,Orbit],
z:z0,
Orbit:[[realpart(z),imagpart(z)]],
for i:1 thru OrbitLength step 1 do
( z:expand(f(z)),
Orbit:endcons([realpart(z),imagpart(z)],Orbit)),
return(Orbit)
)$
/* find fixed points returns a list */
GiveFixedPoints():= block
(
[s],
s:solve(f(z)=z),
/* remove "z=" from list s */
s:map('rhs,s),
s:map('rectform,s),
s:map('float,s),
return(s)
)$
compile(all);
/* ------------ */
iLength:10000;
/* compute a list of critical points */
s:GiveListOfCriticalPoints(f(z));
multiplicities;
length(s);
/* compute orbit of every critical point and join them into one list */
Orbits:[];
for i:1 thru length(s) step 1 do
(
Orbit:GiveOrbit(s[i],iLength),
Orbits:append(Orbit,Orbits)
);
/*-----------------------------------------------------------------------*/
load(draw); /* ( interface to gnuplot ) by Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates */
draw2d(
title = "All critical orbits for f(z)=z^4 -z ",
terminal = png,
user_preamble = "set angles degrees; set grid polar 30; set xtics 0.62996052494744; set mxtics 0.62996052494744; set size square", /* 360/12=30 */
file_name = concat("~/",string(iLength),"b_52"), /* one can put here a path */
pic_width = 1000, /* Since Maxima 5.23, pic_width and pic_height are deprecated. */
pic_height = 1000, /* See option dimensions. To get the same effect, write dimensions=[800,600] */
yrange = [-1.0,1.0],
xrange = [-1.0,1.0],
xlabel = "z.re ",
ylabel = "z.im",
point_type = filled_circle,
points_joined = false,
point_size = 0.7,
key=" critical orbit ",
color =red,
points(Orbits),
point_size = 1.2,
key= "critical points",
color = blue,
points(map(realpart,s),map(imagpart,s)),
key= "fixed parabolic point",
color = black,
points([[0,0]])
);