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

Summary[edit]

Licensing[edit]

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Maxima CAS src code[edit]

• ListOfCriticalPoints : (- 0.7558074500261052 %i) - 0.7558074500261052, 0.2793534499540583 %i - 0.5310341598343944, 0.2793534499540583 - 0.5310341598343943 %i, 0.3674881599064412 %i + 0.3674881599064413 with multiplicities= [1, 1, 1, 1]
• finite fixed points : [(- 0.8 %i) - 0.8, 0.0] with stability: [0.6384000000000008, 1.0]
• distance to fixed points after iLength:30000 forward iterations:
  • d1:GiveDistance(first(Orbit1),fixed[1]) = 0.0624977035289329
  • d2:GiveDistance(first(Orbit2),fixed[2]) = 0.6000296900256318
  • d3:GiveDistance(first(Orbit3),fixed[2]) = 0.6000296900256316
  • d4:GiveDistance(first(Orbit4),fixed[2]) = 0.5197067397512218

batch file for Maxima CAS use : - save as p.mac - run Maxima - batch ("p.mac")

kill(all);
remvalue(all);


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

/* function */
f(z):=z^5 +(0.8+0.8*%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)),
          Orbit:endcons(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)
)$



/* 
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))$

GiveDistance(z1,z2):= cabs(z1-z2)$



compile(all);

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

iLength:30000;


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



fixed :GiveFixedPoints();


stability_list : map(stability, fixed);




Orbit1: GiveOrbit(ListOfCriticalPoints[1],iLength)$
Orbit2: GiveOrbit(ListOfCriticalPoints[2],iLength)$
Orbit3: GiveOrbit(ListOfCriticalPoints[3],iLength)$
Orbit4: GiveOrbit(ListOfCriticalPoints[4],iLength)$

d1: GiveDistance( first(Orbit1), fixed[1]);
d2: GiveDistance( first(Orbit2), fixed[2]);
d3: GiveDistance( first(Orbit3), fixed[2]);
d4: GiveDistance( first(Orbit4), fixed[2]);


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


Orbit1 : ToPoints(Orbit1)$
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+0.8*i)*z^4 + z",
    terminal  = png,
     user_preamble = "set size square", /* 360/26=13.85  ; 360/(2*26)=6,923 */
    file_name = concat("~/Dokumenty/julia_mse/maxima/1/", 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 = false,
    
    
    
    color             = red,
    point_size    = 0.7,
    points_joined = true,
    key=" critical orbit 1",
    Orbit1,
    point_size    = 1.2,
    key= "",
    points_joined = false,
    cr1,
    
    color             = magenta,
    point_size    = 0.7,
    points_joined = true,
    key=" critical orbit 2",
    Orbit2,
    point_size    = 1.4,
    key = "",
    points_joined = false,
    cr2,
    
    color             = grey,
    point_size    = 0.7,
    points_joined = true,
    key=" critical orbit 3",
    Orbit3,
    point_size    = 1.4,
    key= "",
    points_joined = false,
    cr3,
    
    
    color             = purple,
    point_size    = 0.7,
    points_joined = true,
    key=" critical orbit 4",
    Orbit4,
    point_size    = 1.4,
    key= "",
    points_joined = false,
    cr4,
    
    
   
    

    key= "attracting fixed point",
    color           = green,
    f1,	

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

 );

1. ↑ math.stackexchange question: whats-the-difference-between-a-total-basin-of-attraction-and-an-immediate-basin