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Summary

Points at which repelling vectors end :

 vr:solve(B=1);
 vr:map(rhs,vr);
 vr:map(rectform,vr);
 vr:map('float,vr);

Result is a list of 3 complex points :

[0.28867513459481*%i-0.16666666666667,-0.28867513459481*%i-0.16666666666667,0.33333333333333]

with arguments in turns :

[0.33333333333333,0.66666666666667,0.0]

Maxima CAS src code

old code

kill(all);
remvalue(all);

/*------------- functions definitions ---------*/

/* function */
f(z):=z^2 +m*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)
)$

/* riotorto.users.sourceforge.net/gnuplot/vectors/index.html */
GiveAVector(m):=block(
 [x,y,dx,dy],
 x:0,
 y:0,
 dx:realpart(m),
 dy:imagpart(m),
 vector([dx,dy],[-dx,-dy])
)$

GiveRVector(m):=block(
 [x,y,dx,dy],
 x:0,
 y:0,
 dx:realpart(m),
 dy:imagpart(m),
 vector([x,y],[dx,dy])
)$

compile(all);

/* -----const values -------  */

p:1;
q:3;
m:1;
petals:m*q; /* ??? */
/* f^q  */
zq:z;
for i:1 thru q step 1 do zq:f(zq);
zq:expand(zq);
/* Take k term in the expansion of f^q */
k:m*q+1;
kCoeff:coeff(zq,z,k);
/* 
vectors attracting and repelling 
http://math.stackexchange.com/questions/361205/what-is-the-shape-of-parabolic-critical-orbit
*/
B:petals*kCoeff*z^petals;

/* http://math.stackexchange.com/questions/361205/what-is-the-shape-of-parabolic-critical-orbit */
/* attracting vectors */
va:solve(B=-1);
va:map(rhs,va);
va:map(rectform,va);
va:map('float,va);
va:map(GiveAVector,va);

/* repelling vectors */
vr:solve(B=1);
vr:map(rhs,vr);
vr:map(rectform,vr);
vr:map('float,vr);
vr:map(GiveRVector,vr);

/* ------------- main  = computations -----------------*/

/* multiplier of fixed point = coefficient of function f  */
m:float(rectform(exp(2*%pi*%i*p/q)));

iLength:1000;
s:GiveListOfCriticalPoints(f(z));
multiplicities;
length(s);

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 = concat("Critical orbit for f(z)=z^2 + z*",string(m), " where p/q=", string(p/q)),
    terminal  = svg,
     user_preamble = "set size square", /*    */
    file_name = concat("~/maxima/parabolic/critical_orbits/z2plusmz/1over3/arv/",string(iLength),"ss"),
    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 = [-0.6,0.6],
    xrange = [-0.6,0.6],*/
    xlabel     = "z.re ",
    ylabel     = "z.im",

    /* vectors */
    head_both   = false,
    head_length = 0.000001,
    line_width  = 0.3,
    
    head_angle  = 1,
    head_type = nofilled,
    line_type = dots,
    key = "attracting vectors",
    color       = yellow,
    first(va),
    key="",
    rest(va),
    key = "repelling vectors",
    color = gray,
    first(vr),
    key="",
    rest(vr),
    
    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]])

 );

new src code

for 5.38

kill(all);
remvalue(all);

/*------------- functions definitions ---------*/

/* function */
f(z):=z^2 +m*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)
)$

/* riotorto.users.sourceforge.net/gnuplot/vectors/index.html */
GiveAVector(m):=block(
 [x,y,dx,dy],
 x:0,
 y:0,
 dx:realpart(m),
 dy:imagpart(m),
 vector([dx,dy],[-dx,-dy])
)$

GiveRVector(m):=block(
 [x,y,dx,dy],
 x:0,
 y:0,
 dx:realpart(m),
 dy:imagpart(m),
 vector([x,y],[dx,dy])
)$

compile(all);

/* -----const values -------  */

p:1;
q:3;
m:1;
petals:m*q; /* ??? */
/* f^q  */
zq:z;
for i:1 thru q step 1 do zq:f(zq);
zq:expand(zq);
/* Take k term in the expansion of f^q */
k:m*q+1;
kCoeff:coeff(zq,z,k);
/* 
vectors attracting and repelling 
http://math.stackexchange.com/questions/361205/what-is-the-shape-of-parabolic-critical-orbit
*/
B:petals*kCoeff*z^petals;

/* http://math.stackexchange.com/questions/361205/what-is-the-shape-of-parabolic-critical-orbit */
/* attracting vectors */
va:solve(B=-1);
va:map(rhs,va);
va:map(rectform,va);
va:map('float,va);
va:map(GiveAVector,va);

/* repelling vectors */
vr:solve(B=1);
vr:map(rhs,vr);
vr:map(rectform,vr);
vr:map('float,vr);
vr:map(GiveRVector,vr);

/* ------------- main  = computations -----------------*/

/* multiplier of fixed point = coefficient of function f  */
m:float(rectform(exp(2*%pi*%i*p/q)));

iLength:1000;
s:GiveListOfCriticalPoints(f(z));
multiplicities;
length(s);

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 = concat("Critical orbit for f(z)=z^2 + z*",string(m), " where p/q=", string(p/q)),
    terminal  = svg,
    user_preamble = "set size square", /*    */
    file_name = concat("~/maxima/batch/criticalorbit/parabolic/1over3/",string(iLength),"ss"),
    dimensions = [1000,1000],    /* Since Maxima 5.23, pic_width and pic_height are deprecated. */
   /* yrange = [-0.6,0.6],
    xrange = [-0.6,0.6],*/
    xlabel     = "z.re ",
    ylabel     = "z.im",

    /* vectors */
    head_both   = false,
    head_length = 0.000001,
    line_width  = 0.3,
    
    head_angle  = 1,
    head_type = filled,
    line_type = solid,
    line_width    = 5,
    key = "attracting vectors",
    color       = yellow,
    first(va),
    key="",
    rest(va),
    key = "repelling vectors",
    color = gray,
    first(vr),
    key="",
    rest(vr),
    
    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]])

 );