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https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue

Maxiam CAS src code

It is a batch file. Save it as a c.mac and use :

batch("c.mac");

Here is the code :

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]])

 );

References

1. ↑ math.stackexchange question: petals-for-points-near-the-origin