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File:Critical orbit 3d.png

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Summary

Description
English: 3D view of critical orbit of c = i*0.21687214+0.37496784 for complex quadratic polynomial. It tends to weakly attracting fixed point zf with abs(multiplier(zf)=0.99993612384259 . Point c is near root of period 6 component of Mandelbrot set.
Polski: Trójwymiarowy widok orbity punktu krytycznego dla fc(z)=z*z+c. Punkt c jest położonego tuż przy granicy zbioru Mandelbrota. Orbita punktu krytycznego dąży do słabo przyciągającego punktu stałego.
Date
Source

Own work by uploader in Maxima and Gnuplot

 
This plot was created with Gnuplot by n.
Author Adam majewski
Other versions

Long description

This image shows how changes orbit of critical point

for complex quadratic polynomial

Here parameter is constant :

It is in period 1 component, near root of period 6 component of Mandelbrot set.

Axes of three dimensional Cartesian coordinate system :

  • axis x is real part of complex variable
  • axis y is imaginary part of complex variable
  • axis z is number of iteration ( integer number )

Note that axis z is different thing that complex variable

XY complex plane is dynamical plane of complex quadratic polynomial.

Iterations :

  • ( blue point )
  • ( red point)
  • ( red point)
  • ...
  • ( red point)

This image showes that orbit of critical point tends to weakly attracting fixed point.

Maxima source code

/*  
this is batch file for Maxima  5.13.0
http://maxima.sourceforge.net/
tested in wxMaxima 0.7.1
using draw package ( interface to gnuplot ) to draw on the screen
draws  critical orbit = orbit of critical point
*/
c:%i*0.21687214+0.37496784;
/* define function ( map) for dynamical system z(n+1)=f(zn,c)  */
f(z,c):=expand (z*z + c); /* expand speed up  computations and fix the stack overflow problem. Robert Dodier */
/* maximal number of iterations */
iMax:100000; /* to big couses bind stack overflow */
EscapeRadius:10;
/* define z-plane ( dynamical ) */
zxMin:-0.8;
zxMax:0.2;
zyMin:-0.2;
zyMax:0.8;
/* resolution is proportional to number of details and time of drawing */
iXmax:2000;
iYmax:1000;
/* compute critical point */
zcr:rhs(solve(diff(f(z,c),z,1)));
/* save critical point to 2 lists */
xcr:makelist (realpart(zcr), i, 1, 1); /* list of re(z) */
ycr:makelist (imagpart(zcr), i, 1, 1); /* list of im(z) */	
/* ------------------- compute forward orbit of critical point ----------*/
z:zcr; /* first point  */
orbit:[z];
for i:1 thru iMax step 1 do
block
(
 z:f(z,c),
 if abs(z)>EscapeRadius then return(i) else orbit:endcons(z,orbit)
 );
/*-------------- save orbit to draw it later on the screen ----------------------------- */
/* save the z values to 2 lists */
xx:makelist (realpart(f(zcr,c)), i, 1, 1); /* list of re(z) */
yy:makelist (imagpart(f(zcr,c)), i, 1, 1); /* list of im(z) */
zz:makelist (1, i, 1, 1); /* list of iterations */
for i:2 thru length(orbit) step 1 do
block
(
xx:cons(realpart(orbit[i]),xx), 
yy:cons(imagpart(orbit[i]),yy),
zz:cons(i,zz)
);		
/* drawing procedures */
load(draw);/*  draw package by Mario Rodriguez Riotorto http://riotorto.users.sourceforge.net/gnuplot/ archive copy at the Wayback Machine */
draw3d(  
 file_name = "critical_orbit_3d",
 terminal  = 'png,
 pic_width  = iXmax,
 pic_height = iYmax,
 columns  = 1,
 title= concat(""),
 user_preamble = "set grid",
 xlabel     = "Z.re ",
 ylabel     = "Z.im",
 zlabel	="iteration",
 point_type    = filled_circle,
 /*key = "critical point",*/
 color		  =blue,
 points_joined = false,
 points(xcr,ycr,[0]),
 points_joined = false,
 color		  =red,
 point_size    = 0.5,
 points(xx,yy,zz)
 );

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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.
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current10:07, 18 January 2009Thumbnail for version as of 10:07, 18 January 20092,000 × 1,000 (19 KB)Soul windsurfer{{Information |Description={{en|1=3D view of critical orbit of c:%i*0.21687214+0.37496784 for complex quadratic polynomial. It tends to weakly attracting point near root of period 6 component.}} {{pl|1=Trójwymiarowy widok orbity punktu krytycznego dla fc

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