Summary

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Long description

Definition of hyperbolic components ( system of 2 equations)

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

Solving system of equations

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 archive copy at the 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]

Result of solving

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

Drawing

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

Set of points looks like curve.

Maxima source code

/* 
 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("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)
 );

See also

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

References

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). archive copy at the Wayback Machine
5. ↑ jtroot3 Maxima package by Raymond Toy archive copy at the 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 archive copy at the Wayback Machine

Acknowledgements

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

Licensing

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