File:Parabolic critical orbit for internal angle one fifth.png
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Summary
DescriptionParabolic critical orbit for internal angle one fifth.png |
English: parabolic critical orbit for internal angle one fifth for fc(z) = z^2 + c |
Date | |
Source | Own work |
Author | Adam majewski |
Licensing
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See also
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Critical Orbit, Inner and outer circle for Golden Mean Quadratic Julia set
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Critical orbit tends to period 3 orbit
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3D view of critical orbit tending to parabolic fixed point
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distance between points of critical orbit in case of attracting fixed point
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Distance to fixed point for various types of dynamics
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Siegel disc
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parabolic cases
Maxima CAS src code
/* this is batch file for Maxima 5.22.1 http://maxima.sourceforge.net/ tested in wxMaxima wxMaxima 0.8.5 using draw package ( interface to gnuplot ) to draw on the screen */ kill(all); /* ---------- functions ---------------------- */ /* conformal map from circle to cardioid ( boundary of period 1 component of Mandelbrot set */ F(w):=w/2-w*w/4; /* circle D={w:abs(w)=1 } where w=l(t,r) t is angle in turns ; 1 turn = 360 degree = 2*Pi radians r is a radius */ l(t,r):=r*%e^(%i*t*2*%pi); /* http://en.wikipedia.org/wiki/Complex_quadratic_polynomial */ f(z,c):=z*z+c $ GiveCriticalOrbit(c,iMax,denominator):= /* computes (without escape test) critical orbit (forward orbit of critical point ) and saves it to the list */ block( [z,orbit], z:0, /* first point = critical point z:0+0*%i */ orbit:[z], for i:1 thru iMax step 1 do ( z:expand(f(z,c)), orbit:endcons(z,orbit)), return(orbit) )$ /* find fixed point alfa */ GiveFixed(c):= float(rectform((1-sqrt(1-4*c))/2))$ /* give angle with respect to alfa */ GiveAngle(z,alfa):= carg(z-alfa) $ compile(all)$ /* ---------- constant ---------------------------*/ Numerator :1; denominator :5; internalAngle: Numerator/denominator; internalRadius:1; iMax:50000000; /* -------------- main ----------------- */ /* point of unit circle w:l(internalAngle,internalRadius); */ w1:l(internalAngle,internalRadius); /* point of circle */ c: float(rectform(F(w1))) ; /* point of period 1 component of Mandelbrot set */ zAlfa:GiveFixed(c); /* ------------------- compute 5 subsets of forward orbit of critical point ----------*/ z:0; /* first point */ orbit1:[z]; for i:1 thru (denominator-1) step 1 do ( z:rectform(f(z,c)), if i=1 then orbit2:[z] elseif i=2 then orbit3:[z] elseif i=3 then orbit4:[z] elseif i=4 then orbit5:[z] ); for i:(denominator) thru iMax step 1 do ( z:rectform(f(z,c)), j:mod(i,denominator), 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) ); load(draw); /* ( interface to gnuplot ) by Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates */ draw2d( title= " Critical Orbit in parabolic case for internal angle = 1/5 ", user_preamble = "", terminal = 'png, pic_width = 1000, pic_height = 1000, xlabel = "Z.re ", ylabel = "Z.im", point_type = filled_circle, points_joined = true, point_size = 1.5, /* */ color =red, key = "1", points(map(realpart,orbit1),map(imagpart,orbit1)), /* */ color =green, key = "2", points(map(realpart,orbit2),map(imagpart,orbit2)), /* */ color =blue, key = "3", points(map(realpart,orbit3),map(imagpart,orbit3)), /* */ color =yellow, key = "4", points(map(realpart,orbit4),map(imagpart,orbit4)), /* */ color =magenta, key = "5", points(map(realpart,orbit5),map(imagpart,orbit5)), /* */ color =black, key = "parabolic fixed point", points([realpart(zAlfa)],[imagpart(zAlfa)]), /* */ color =brown, key = "critical point", points([0],[0]) );
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14 January 2012
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 21:46, 14 January 2012 | 1,000 × 1,000 (24 KB) | Soul windsurfer |
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