Critical_orbit_3d.png
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Summary
Description Critical 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.
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Date | |
Source |
Own work by uploader in Maxima and Gnuplot
This plot was created with
Gnuplot
.
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Author | Adam majewski |
Other versions |
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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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