File:Critical orbit f(z) = z*z+c and c=-0.749413589136570+0.015312826507689*i.png

Summary[edit]

 

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Summary[edit]

// program by marcm200

coefficients read from input file spiral1.txt
	degree 2 coefficient = ( +1.0000000000000000 +0.0000000000000000*i) 
	degree 0 coefficient = ( -0.7494135891365700 +0.0153128265076890*i) 

Input polynomial p(z)=(1+0i)*z^2+(-0.74941358913656996865+0.015312826507689000083i)

1 critical points found

	cp#0: 0,0 . It's critical orbit is bounded and enters cycle #0 length=1 and it's stability = |multiplier|=0.99959  
	internal angle = 0.49756114816558294489
cycle = {
-0.49973608562614335593,0.0076584343005330103582 ; }

// m-describe by Claude 
the input point was -7.4941358913656997e-01 + +1.5312826507689e-02 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
  with nucleus at +0e+00 + +0e+00 i
  the component has size 1.00000e+00 and is pointing west
  the atom domain has size 0.00000e+00
  the atom domain coordinates of the input point are -nan + -nan i
  the atom domain coordinates in polar form are nan to the east
  the nucleus is 7.49570e-01 to the east of the input point
  the input point is interior to this component at
  radius 9.99590e-01 and angle 0.497561148125481134 (in turns)
  the multiplier is -9.99472e-01 + +1.53169e-02 i
  a point in the attractor is -4.99741e-01 + +7.65848e-03 i
  external angles of this component are:
  .(0)
  .(1)

Maxima CAS src code[edit]

kill(all);
remvalue(all);

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

/* function */
f(z):=z^2 + c;

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:[z0], 
 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))]$

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


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



compile(all);

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

c: -0.749413589136570  +0.015312826507689*%i$
zcr:0.0$
iLength:10000;
/* ------------- main  = computations -----------------*/

 /* compute fixed points */
 Beta:float(rectform((1+sqrt(1-4*c))/2))$ /* compute repelling fixed point beta */
 alfa:float(rectform((1-sqrt(1-4*c))/2))$ /* other fixed point */



 Orbit:GiveOrbit(zcr,iLength)$
 


Orbit:ToPoints(Orbit)$
zcr:ToPoint(zcr)$

alfa:ToPoint(alfa)$

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


path:"~/Dokumenty/construct/2/pauldebrot1/"$  /*  if empty then file is in a home dir */

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 +", string(c)),
    terminal  = png,
    user_preamble = "set size square", /*    */
    file_name = concat(path ,string(iLength),"_8"),
    dimensions = [1500,1500],    /* Since Maxima 5.23, pic_width and pic_height are deprecated. */
    xrange = [-0.8,0.0],
    yrange = [-0.4,0.4],
    xlabel     = "z.re ",
    ylabel     = "z.im",

    
    
    point_type    = 7, 
    points_joined = false,
    point_size    = 0.8,
    key=" critical orbit ",
    color             =red,
    Orbit,
   
    point_size    = 1.2,
    key= "critical point",
    color           = blue,
    zcr,
    
    

    key= "fixed point",
    color           = black,
    alfa

 );