Hi,
Thx for all posts.
This image shows how c changes on parameter plane ( escape route )
in image http://commons.wikimedia.org/w/index.php?title=Image:6furcation.gif
which shows dynamic plane ( z-plane)
Adam
/*
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
by Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates
draws escape route for angle 1/6 :
- from center of period 1 component to root point of period 6 component
( along 1/6 internal ray of period 1 component)
-from root point of period 6 component to center of period 6 component
( along angle 0 internal ray of period 6 component)
- from center of period 6 component to root point of period 36 component
( along angle 1/6 internal ray of period 6 component)
.... ( this is endless journey ) ....
*/
/* center of main cardioid */
Cx1c:makelist (0, i, 1, 1);
Cy1c:makelist (0, i, 1, 1);
/* root point of period 6 component */
radius:1;
angle:1/6; /* in radians */
l:radius * exp(%i*angle*2*%pi);
C6r:l/2 - (l*l)/4;
trigsimp(%);
rectform(%);
C6r:float(%), numer;
Cx6r:makelist (realpart(C6r), i, 1, 1);
Cy6r:makelist (imagpart(C6r), i, 1, 1);
/* center of period 6 component */
C6c:0.38900684056977+0.21585065087082*%i;
Cx6c:makelist (realpart(C6c), i, 1, 1);
Cy6c:makelist (imagpart(C6c), i, 1, 1);
/* radius of period 6 component */
r:abs(C6c-C6r);
/* line coefficient */
d6x0:realpart(C6c)-realpart(C6r);
d6y0:imagpart(C6c)-imagpart(C6r);
/* drawing code */
load(draw)$
draw2d (
title="1/6 escape route in Mandelbrot set ",
user_preamble = "set key outside top",
xrange = [ 0.00, 0.415],
yrange = [0.00,0.415],
nticks = 200,
/* curves */
key ="period 1 component",
parametric( cos(t)/2-cos(2*t)/4, sin(t)/2-sin(2*t)/4, t,-%pi,%pi),
color =red,
key ="internal ray for angle 1/6",
parametric(t^2/8+t/4,((sqrt(3)*t)/4-(sqrt(3)*t^2)/8),t,0,1),
color =black,
key ="period 2 component",
parametric( cos(t)/4-1, sin(t)/4, t,-%pi,%pi),
color =magenta,
key ="period 6 component",
parametric( realpart(C6c)+r*cos(t), imagpart(C6c)+r*sin(t), t,-%pi,%pi),
color =purple,
key ="internal ray for angle 0",
parametric(realpart(C6r)+t*d6x0 ,imagpart(C6r)+t*d6y0,t,0,1),/* */
/* points */
point_type = filled_circle,
color =blue,
point_size = 0.5,
key ="center of period 1 component",
points(Cx1c,Cy1c),
color =brown,
key ="root of period 6 component",
points(Cx6r,Cy6r),
color =green,
key ="center of period 6 component",
points(Cx6c,Cy6c)
)$