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Boundaries of hyperbolic components for period n of Mandelbrot set are defined by system of equations^[1] :

${\displaystyle {\begin{cases}F(n,z,c)=z\\{\frac {dF(n,z,c)}{dz}}=w\end{cases}}}$

Above system of 2 equations has 3 variables : ${\displaystyle z,c,w\,}$ ( n is constant). One have to remove 1 variable to be able to solve it.

Boundaries are closed curves : cardioids or circles. One can parametrize points of boundaries with angle ${\displaystyle t\,}$ ( here measured in turns from 0 to 1 ).

After evaluation of ${\displaystyle w=l(t)\,}$ one can put it into above system, and get a system of 2 equations with 2 variables ${\displaystyle z,c\,}$.

Now it can be solved

Method of solving system of equation :^[2]

• Newton method for solving system of nonlinear equations in more then one variable (Maxima implementation = function mnewton^[3] )
• centers of hyperbolic component as an initial approximation

Using Newton method is based on Mark McClure kopia archiwizowana w Wayback Machine's paper "Bifurcation sets and critical curves"^[4]

Computing centers of hyperbolic components for given period n:

• compute center for given period n ( Maxima function polyroots^[5][6] or allroots ^[7])
• remove centers for dividers of n. It can be done by dividing polynomials ( Robert Munafo method)^[8]

Solving above system gives one point c of each hyperbolic compponent of period n for each angle t ( point w ). Together it gives a list of points

Draw a list of points ( on the sceen or to the file using Maxima draw2d function ^[9])

Set of points looks like curve.

 /* 
 batch file for Maxima
 http://maxima.sourceforge.net/
 wxMaxima 0.7.6 http://wxmaxima.sourceforge.net
 Maxima 5.16.1 http://maxima.sourceforge.net
 Using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
 Distributed under the GNU Public License. 
 based on :
 Mark McClure "Bifurcation sets and critical curves" - Mathematica in Education and Research, Volume 11, issue 1 (2006).
 */

 start:elapsed_run_time ();

 load("mnewton")$
 newtonepsilon: 1.e-3;
 newtonmaxiter: 100;
 load("C:/Program Files/Maxima-5.13.0/share/maxima/5.13.0/share/polyroots/jtroot3.mac")$ /*  Raymond Toy http://common-lisp.net/~rtoy/jtroot3.mac */
 maperror:false;
 fpprec : 16;

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

 /* basic funtion */
 f(z,c):=z*z+c$
 /* */
 F(n, z, c) :=
 if n=1 then f(z,c)
 else f(F(n-1, z, c),c)$
 
/* */
 G(n,z,c):=F(n, z, c)-z$
 iMax:100; /* number of points to draw */
 dt:1/iMax;

 /* 
 unit circle D={w:abs(w)=1 } where w=l(t) 
 t is angle in turns ; 1 turn = 360 degree = 2*Pi radians 
 */
 l(t):=%e^(%i*t*2*%pi);
 

/* point to point method of drawing */
 t:0; /* angle in turns */ 
 /* compute first point of curve, create list and save point to this list */
 /* point of unit circle   w:l(t); */
 w:rectform(ev(l(t), numer)); /* "exponential form prevents allroots from working", code by Robert P. Munafo */ 

 /* period 1 */
 p:1;
 /*   center of component */
 ec:G(p,0,c)$
 center1:polyroots(ec,c);
 nMax1:length(center1);
 /* boundary point */
 e11:expand(G(p,z,c))$
 e12:expand(diff(F(p,z,c),z))$
 c1:mnewton([e11, e12-w], [z,c], [center1[1], center1[1]]);  /* code by Robert P. Munafo  */
 nMax1:length(c1);
 xx1:makelist (realpart(rhs(c1[1][2])), i, 1, 1); 
 yy1:makelist (imagpart(rhs(c1[1][2])), i, 1, 1); 

 /* period 2 */
 p:2;
 /*   center of component */
 ec:radcan(G(p,0,c)/G(1,0,c))$ /* code by Robert P. Munafo  and all similar beyond  */
 center2:polyroots(ec,c);
 nMax2:length(center2);
 /* boundary point */
 e21:radcan(G(p,z,c)/G(1,z,c))$
 e22:expand(diff(F(p,z,c),z))$
 c2:mnewton([e21, e22-w], [z,c], [center2[1], center2[1]]);
 xx2:makelist (realpart(rhs(c2[1][2])), i, 1, 1); 
 yy2:makelist (imagpart(rhs(c2[1][2])), i, 1, 1);  

 /* period 3 */
 p:3;
 /*   center of component */
 ec:radcan(G(p,0,c)/G(1,0,c))$
 center3:polyroots(ec,c);
 nMax3:length(center3);
 /* boundary point */
 e31:radcan(G(p,z,c)/G(1,z,c))$
 e32:expand(diff(F(p,z,c),z))$
 /*  */
 c3:mnewton([e31, e32-w], [z,c], [center3[1], center3[1]]);
 xx3:makelist (realpart(rhs(c3[1][2])), i, 1, 1); 
 yy3:makelist (imagpart(rhs(c3[1][2])), i, 1, 1); 
 for n:2 thru nMax3 step 1 do /* all components in 1 list */
 block
 ( 
  c3:mnewton([e31, e32-w], [z,c], [center3[n], center3[n]]),
  xx3:cons(realpart(rhs(c3[1][2])),xx3),
  yy3:cons(imagpart(rhs(c3[1][2])),yy3)
 );

 /* period 4 */
 /*   center of component */
 ec:radcan(G(4,0,c)/G(2,0,c))$
 center4:polyroots(ec,c);
 nMax4:length(center4);
 /* boundary point */
 e41:radcan(G(4,z,c)/G(2,z,c))$
 e42:expand(diff(F(4,z,c),z))$
 c4:mnewton([e41, e42-w], [z,c], [center4[1], center4[1]]);
 xx4:makelist (realpart(rhs(c4[1][2])), i, 1, 1); 
 yy4:makelist (imagpart(rhs(c4[1][2])), i, 1, 1); 
 for n:2 thru nMax4 step 1 do /* all components in 1 list */
 block
 ( 
  c4:mnewton([e41, e42-w], [z,c], [center4[n], center4[n]]),
  xx4:cons(realpart(rhs(c4[1][2])),xx4),
  yy4:cons(imagpart(rhs(c4[1][2])),yy4)
 ); 

 /* period 5 */
 newtonmaxiter: 200;
 /*   center of component */
 ec:radcan(G(5,0,c)/G(1,0,c))$
 center5:polyroots(ec,c);
 nMax5:length(center5);
 /* boundary point */ 
 e51:radcan(G(5,z,c)/G(1,z,c))$
 e52:expand(diff(F(5,z,c),z))$ 
 c5:mnewton([e51, e52-w], [z,c], [center5[1], center5[1]]);
 xx5:makelist (realpart(rhs(c5[1][2])), i, 1, 1); 
 yy5:makelist (imagpart(rhs(c5[1][2])), i, 1, 1); 
 for n:2 thru nMax5 step 1 do /* all components in 1 list */
 block
 ( 
  c5:mnewton([e51, e52-w], [z,c], [center5[n], center5[n]]),
  xx5:cons(realpart(rhs(c5[1][2])),xx5),
  yy5:cons(imagpart(rhs(c5[1][2])),yy5)
 ); 

 /* ------------*/
 for i:1 thru iMax step 1 do
 block
 ( 
 t:t+dt,
 w:rectform(ev(l(t), numer)), /* "exponential form prevents allroots from working", code by Robert P. Munafo */ 
 /* period 1 */
 c1:mnewton([e11, e12-w], [z,c], [center1[1], center1[1]]),
 xx1:cons(realpart(rhs(c1[1][2])),xx1),
 yy1:cons(imagpart(rhs(c1[1][2])),yy1),
 /* period 2 */
 c2:mnewton([e21, e22-w], [z,c], [center2[1], center2[1]]),
 xx2:cons(realpart(rhs(c2[1][2])),xx2),
 yy2:cons(imagpart(rhs(c2[1][2])),yy2),
 /* period 3*/
 for n:1 thru nMax3 step 1 do 
 block
 (	c3:mnewton([e31, e32-w], [z,c], [center3[n], center3[n]]),
  xx3:cons(realpart(rhs(c3[1][2])),xx3),
  yy3:cons(imagpart(rhs(c3[1][2])),yy3)
 ),
 /* period 4 */
 if evenp(i) then
 for n:1 thru nMax4 step 1 do 
 block
 (  	c4:mnewton([e41, e42-w], [z,c], [center4[n], center4[n]]),
 xx4:cons(realpart(rhs(c4[1][2])),xx4),
 yy4:cons(imagpart(rhs(c4[1][2])),yy4)
 ),
 /* period 5 */
 if evenp(i) then
 for n:1 thru nMax5 step 1 do /* all components in 1 list */
 block
  (  	c5:mnewton([e51, e52-w], [z,c], [center5[n], center5[n]]),
  xx5:cons(realpart(rhs(c5[1][2])),xx5),
  yy5:cons(imagpart(rhs(c5[1][2])),yy5)
  )
 );

 stop:elapsed_run_time ();
 time:fix(stop-start); 
 nMax:nMax1+nMax2+nMax3+nMax4+nMax5;

 load(draw);

 draw2d(
   file_name = "c4n", /* file in directory  C:\Program Files\Maxima-5.16.1\wxMaxima */
   terminal  = 'jpg, /* jpg when draw to file with jpg extension */
   pic_width  = 1000,
   pic_height = 1000,
   yrange = [-1.5,1.5],
   xrange = [-2,1],
   title= concat("Boundaries of ",string(nMax)," hyperbolic components of Mandelbrot set in ",string(time)," sec"),
   xlabel     = "c.re ",
   ylabel     = "c.im",
   point_type    = dot,
   point_size    = 5,
   points_joined =true,
   user_preamble="set size square;set key out vert;set key bot center",
   color = black,
   key = "one period 1 component  ",
   points(xx1,yy1),
   key = "one period 2 component  ",
   color         = green,
   points(xx2,yy2),
   points_joined =false,
   color         = red,
   key = concat(string(nMax3)," period 3 components  "),
   points(xx3,yy3),
   key = concat(string(nMax4)," period 4 components "),
   points(xx4,yy4),
   key = concat(string(nMax5)," period 5 components "),
   points(xx5,yy5)
 );

• Other method of drawing based on boundary equations
• Centers of hyperbolic components computed with MPSolve

1. ↑ WikiBooks/Fractals/Iterations in the complex plane/Mandelbrot set
2. ↑ Robert P. Munafo - private communcation
3. ↑ Maxima Manual: 63. mnewton
4. ↑ Mark McClure "Bifurcation sets and critical curves" - Mathematica in Education and Research, Volume 11, issue 1 (2006). kopia archiwizowana w Wayback Machine
5. ↑ jtroot3 Maxima package by Raymond Toy kopia archiwizowana w Wayback Machine
6. ↑ cvs /maxima/share/numeric/jtroot3.mac
7. ↑ Maxima Manual: 21. function allroots
8. ↑ Robert P. Munafo - private communcation
9. ↑ Maxima draw package by Mario Rodríguez Riotorto kopia archiwizowana w Wayback Machine

This program is not only my work but was done with help of many great people (see references). Warm thanks (:-))

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