Jump to content

File:ComponentNewton.jpg

Page contents not supported in other languages.
This is a file from the Wikimedia Commons
From Wikipedia, the free encyclopedia

Original file (1,000 × 1,000 pixels, file size: 58 KB, MIME type: image/jpeg)

This file has been superseded by Mandelbrot set Component by Newton method.png. It is recommended to use the other file. Please note that deleting superseded images requires consent.
new file

Summary

Description
English: Boundaries of Components of Mandelbrot set by Newton method
Polski: Brzeg składowych zbioru Mandelbrot obliczony metodą Newtona
Date
Source Own work by uploader ( using Maxima and Gnuplot ) with help of many people ( see references )
Author Adam majewski

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] :

Above system of 2 equations has 3 variables : ( 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 ( here measured in turns from 0 to 1 ).

After evaluation of one can put it into above system, and get a system of 2 equations with 2 variables .

Now it can be solved

Solving system of equations

Method of solving system of equation :[2]

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.

 /* 
 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)
 );

See also

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

I, the copyright holder of this work, hereby publish it under the following licenses:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
You may select the license of your choice.

Captions

Add a one-line explanation of what this file represents

Items portrayed in this file

depicts

26 September 2008

image/jpeg

3b3138087aa5981d78362ded9a55c979d4b31d6e

59,769 byte

1,000 pixel

1,000 pixel

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current17:54, 27 September 2008Thumbnail for version as of 17:54, 27 September 20081,000 × 1,000 (58 KB)Soul windsurfer{{Information |Description= |Source= |Date= |Author= |Permission= |other_versions= }}
16:58, 26 September 2008Thumbnail for version as of 16:58, 26 September 20081,000 × 1,000 (56 KB)Soul windsurfer{{Information |Description={{en|1=Boundaries of Components of Mandelbrot set by Newton method}} |Source=Own work by uploader |Author=Adam majewski |Date=26.09.2008 |Permission= |other_versions= }} Components of Mandelbrot set by New
No pages on the English Wikipedia use this file (pages on other projects are not listed).

Global file usage

The following other wikis use this file:

Metadata