Summary

DescriptionCritical orbit 3d.png	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	17 January 2009
Source	Own work by uploader in Maxima and Gnuplot   This plot was created with Gnuplot.
Author	Adam majewski
Other versions	2D view with 6 colors

Long description

This image shows how changes orbit of critical point

${\displaystyle z_{cr}=0\,}$

for complex quadratic polynomial

${\displaystyle f_{c}(z)=z*z+c\,}$

Here parameter ${\displaystyle c\,}$ is constant :

${\displaystyle c=0.37496784+i*0.21687214\,}$

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 ${\displaystyle z\,}$
• axis y is imaginary part of complex variable ${\displaystyle z\,}$
• axis z is number of iteration ( integer number )

Note that axis z is different thing that complex variable ${\displaystyle z\,}$

XY complex plane is dynamical plane of complex quadratic polynomial.

Iterations :

• ${\displaystyle z_{0}=z_{cr}=0\,}$ ( blue point )
• ${\displaystyle z_{1}=f_{c}(z_{0})=c\,}$ ( red point)
• ${\displaystyle z_{2}=f_{c}(z_{1})=c^{2}+c\,}$ ( red point)
• ...
• ${\displaystyle z_{n+1}=f_{c}(z_{n})\,}$ ( 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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