File:Cr6spiral3d.png
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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.
Summary
editDescriptionCr6spiral3d.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. |
Date | |
Source | Own work |
Author | Adam majewski |
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![](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Lissajous_small.svg/21px-Lissajous_small.svg.png)
This plot was created with Gnuplot by n.
Maxima CAS src code
edit/* 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 Adam Majewski fraktal.republika.pl with help of Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates */ /* */ 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:1000; 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 attracting fixed point */ fixed:solve([z=f(z,c)], [z]); zfa:if cabs(2*rhs(fixed[1]))<1 then rhs(fixed[1]) else rhs(fixed[2]); /* check multipliers of fixed points */ float(cabs(2*rhs(fixed[1]))); float(cabs(2*rhs(fixed[2]))); /* save attracting fixed point to 2 lists */ xfa:makelist (realpart(zfa), i, 1, 1); /* list of re(z) */ yfa:makelist (imagpart(zfa), i, 1, 1); /* list of im(z) */ /* ------------------- compute forward orbit of critical point in 6 subsets ----------*/ z:zcr; /* first point */ for i:1 thru 6 step 1 do block ( z:f(z,c), if i=1 then orbit1:[z] elseif i=2 then orbit2:[z] elseif i=3 then orbit3:[z] elseif i=4 then orbit4:[z] elseif i=5 then orbit5:[z] elseif i=6 then orbit6:[z] ); for i:6 thru iMax step 1 do block ( z:f(z,c), if abs(z)>EscapeRadius then exit(i) else ( j:mod(i,6), if j=0 then orbit1:endcons(z,orbit1) elseif j=1 then orbit2:endcons(z,orbit2) elseif j=2 then orbit3:endcons(z,orbit3) elseif j=3 then orbit4:endcons(z,orbit4) elseif j=4 then orbit5:endcons(z,orbit5) elseif j=5 then orbit6:endcons(z,orbit6) ) ); z1:makelist(i*6,i,1,length(orbit1)); z2:makelist((i+1)*6,i,1,length(orbit2)); z3:makelist((i+2)*6,i,1,length(orbit3))$ z4:makelist((i+3)*6,i,1,length(orbit4))$ z5:makelist((i+4)*6,i,1,length(orbit5))$ z6:makelist((i+5)*6,i,1,length(orbit6))$ /* draw reversed orbit of beta using draw package */ load(draw); draw3d( title= concat(" dynamical plane for f(z,c):=z*z+",string(c)), user_preamble = "set size square;set key out;set key top right", file_name = "cr_6spiral_3d", terminal = 'png, pic_width = iXmax, pic_height = iYmax, /* yrange = [zyMin,zyMax], xrange = [zxMin,zxMax], */ xlabel = "Z.re ", ylabel = "Z.im", point_type = filled_circle, points_joined = false, point_size = 0.8, key = "critical point", color =magenta, points(xcr,ycr,[0]), point_size = 0.2, color =red, key = "1-spiral of critical orbit", points(map (realpart,orbit1),map (imagpart,orbit1),z1), color =orange, key = "2-spiral of critical orbit", points(map (realpart,orbit2),map (imagpart,orbit2),z2), color =yellow, key = "3-spiral of critical orbit", points(map (realpart,orbit3),map (imagpart,orbit3),z3), color =green, key = "4-spiral of critical orbit", points(map (realpart,orbit4),map (imagpart,orbit4),z4), color =cyan, key = "5-spiral of critical orbit", points(map (realpart,orbit5),map (imagpart,orbit5),z5), color =blue, key = "6-spiral of critical orbit", points(map (realpart,orbit6),map (imagpart,orbit6),z6), point_size = 0.8, key = "attracting fixed point", color =black, points(xfa,yfa,[iMax+1]) );
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