File:Critical orbits f(z) =z^5 +(0.8+0.4)*z^4 + z.svg

Summary[edit]

Licensing[edit]

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Maxima CAS src code[edit]

• ListOfCriticalPoints = [(- 0.5574028094466224 %i) - 0.7341195567603783, 0.3241998748996613 %i - 0.5938111474337146, 0.3239311106009048 - 0.4918402540438871 %i, 0.4050431885908482 %i + 0.3639995935931879]
• fixed points : [(- 0.4 %i) - 0.8, 0.0] with stability_list = [1.025280449438104, 1.0]
• (- 0.5574028094466224 %i) - 0.7341195567603783 is escaping
• 0.3241998748996613 %i - 0.5938111474337146 is not escaping
• 0.3239311106009048 - 0.4918402540438871 %i is not escaping
• 0.4050431885908482 %i + 0.3639995935931879 orbit is not escaping

/*



*/


kill(all);
remvalue(all);


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

/* function */
f(z):=z^5 +(0.8+0.4*%i)*z^4 + z;


/* multiplier */
define(multiplier(z), diff(f(z),z,1));


stability(z):=
block(
	[s],
	s:multiplier(z),
	s: rectform(s),
	s: float(s),
	s: cabs(s),
	return(s)
)$




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:[z], 
 for i:1 thru OrbitLength step 1 do
        ( z:expand(f(z)),
         if cabs(z) > 2 then 
         	( print(z0), print("orbit of is escaping"),  
         	  return(Orbit)),
          Orbit:endcons(z ,Orbit)
          ),
 print(z0), print("orbit is not escaping"),  
 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)
)$



/* 
converts complex number z = x*y*%i 
to the list in a draw format:  
[x,y] 
*/
d(z):=[float(realpart(z)), float(imagpart(z))]$

ToPoint(z):= points([d(z)])$

/* give Draw List from one point*/
dl(z):=[d(z)]$

ToPoints(myList):= points(map(d,myList))$





compile(all);

/* ------------  */

iLength:6000;


ListOfCriticalPoints:GiveListOfCriticalPoints(f(z))$
multiplicities;
ListOfCriticalPoints;



fixed :GiveFixedPoints();


stability_list : map(stability, fixed);


OrbitEscaping: GiveOrbit(ListOfCriticalPoints[1],iLength)$

Orbit2: GiveOrbit(ListOfCriticalPoints[2],iLength)$
Orbit3: GiveOrbit(ListOfCriticalPoints[3],iLength)$
Orbit4: GiveOrbit(ListOfCriticalPoints[4],iLength)$



/* fixed: ToPoints(fixed)$ */
f1: ToPoint(fixed[1]);
f2: ToPoint(fixed[2]);

OrbitEscaping : ToPoints(OrbitEscaping)$
Orbit2 : ToPoints(Orbit2)$
Orbit3 : ToPoints(Orbit3)$
Orbit4 : ToPoints(Orbit4)$


cr1: ToPoint(ListOfCriticalPoints[1])$
cr2: ToPoint(ListOfCriticalPoints[2])$
cr3: ToPoint(ListOfCriticalPoints[3])$
cr4: ToPoint(ListOfCriticalPoints[4])$



/*-----------------------------------------------------------------------*/
load(draw); /* ( interface to gnuplot ) by Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates */

draw2d(
    title = "All  critical orbits for f(z)=z^5 +(0.8+0.4*i)*z^4 + z",
    terminal  = svg,
     user_preamble = "set size square; set key left top;", /* 360/26=13.85  ; 360/(2*26)=6,923 */
    file_name = concat("~/Dokumenty/julia_mse/maxima/04/2/p4_", string(iLength)),
    dimensions  = [1000, 1000],   /* Since Maxima 5.23, pic_width and pic_height are deprecated. */
    /*
    yrange = [-1.0,1.0],
    xrange = [-1.0,1.0],
    */
    xlabel     = "z.re ",
    ylabel     = "z.im",
    
    point_type    = filled_circle,
    points_joined = true,
    
    point_size    = 0.4,
    key="non escaping critical orbit 2",
    color             =light_red ,
    Orbit2,
    point_size    = 1.0,
    key= "critical point 2",
    points_joined = false,
    color           = red,
    cr2,
    
    
    
    key="non escaping critical orbit 3",
    point_size    = 0.4,
    points_joined = true,
    color             =light_blue ,
    Orbit3,
    point_size    = 1.0,
    key= "critical point 3",
    points_joined = false,
    color           = blue,
    cr3,
    
    
    
    key="non escaping critical orbit 4",
    point_size    = 0.4,
    points_joined = true,
    color             =light_green ,
    Orbit4,
    point_size    = 1.0,
    key= "critical point 4",
    points_joined = false,
    color           = green,
    cr4,
    
    
    points_joined = false,
    point_size    = 0.4,
    key="escaping critical orbits",
    color             = light_grey, /* https://maxima.sourceforge.io/docs/manual/maxima_218.html#Item_003a-draw_002fdefvr_002fcolor */
    OrbitEscaping,

    point_size    = 1.0,
    key= "escaping critical point 1",
    color           = light_grey,
    cr1,
    
    point_size    = 1.5,
    point_type    = 9,	
    key= "repelling fixed point",
    color           = light_grey,
    f1,	

    key= "parabolic fixed point",
    color           = magenta,
    f2

 );