File:Critical orbit for f(z)=z^2 + mz where p over q=1 over 3.svg
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Captions
DescriptionCritical orbit for f(z)=z^2 + mz where p over q=1 over 3.svg |
English: Critical orbit for f(z)=z^2 + mz where p/q=1/3 |
Date | |
Source | Own work |
Author | Adam majewski |
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Summary[edit]
Points at which repelling vectors end :
vr:solve(B=1); vr:map(rhs,vr); vr:map(rectform,vr); vr:map('float,vr);
Result is a list of 3 complex points :
[0.28867513459481*%i-0.16666666666667,-0.28867513459481*%i-0.16666666666667,0.33333333333333]
with arguments in turns :
[0.33333333333333,0.66666666666667,0.0]
Maxima CAS src code[edit]
old code[edit]
kill(all); remvalue(all); /*------------- functions definitions ---------*/ /* function */ f(z):=z^2 +m*z; GiveListOfCriticalPoints(fun):= block( [d,s], /* derivative */ d:diff(fun,z,1), /* critical points z: d=0 */ s:solve(d=0,z), /* remove "z=" from list s */ s:map('rhs,s), /* convert to form x+y*%i */ s:map('rectform,s), s:map('float,s), return(s) )$ /* f(z) is used as a global function I do not know how to put it as a argument */ GiveOrbit(z0,OrbitLength):= block( [z,Orbit], z:z0, Orbit:[[realpart(z),imagpart(z)]], for i:1 thru OrbitLength step 1 do ( z:expand(f(z)), Orbit:endcons([realpart(z),imagpart(z)],Orbit)), return(Orbit) )$ /* find fixed points returns a list */ GiveFixedPoints():= block ( [s], s:solve(f(z)=z), /* remove "z=" from list s */ s:map('rhs,s), s:map('rectform,s), s:map('float,s), return(s) )$ /* riotorto.users.sourceforge.net/gnuplot/vectors/index.html */ GiveAVector(m):=block( [x,y,dx,dy], x:0, y:0, dx:realpart(m), dy:imagpart(m), vector([dx,dy],[-dx,-dy]) )$ GiveRVector(m):=block( [x,y,dx,dy], x:0, y:0, dx:realpart(m), dy:imagpart(m), vector([x,y],[dx,dy]) )$ compile(all); /* -----const values ------- */ p:1; q:3; m:1; petals:m*q; /* ??? */ /* f^q */ zq:z; for i:1 thru q step 1 do zq:f(zq); zq:expand(zq); /* Take k term in the expansion of f^q */ k:m*q+1; kCoeff:coeff(zq,z,k); /* vectors attracting and repelling http://math.stackexchange.com/questions/361205/what-is-the-shape-of-parabolic-critical-orbit */ B:petals*kCoeff*z^petals; /* http://math.stackexchange.com/questions/361205/what-is-the-shape-of-parabolic-critical-orbit */ /* attracting vectors */ va:solve(B=-1); va:map(rhs,va); va:map(rectform,va); va:map('float,va); va:map(GiveAVector,va); /* repelling vectors */ vr:solve(B=1); vr:map(rhs,vr); vr:map(rectform,vr); vr:map('float,vr); vr:map(GiveRVector,vr); /* ------------- main = computations -----------------*/ /* multiplier of fixed point = coefficient of function f */ m:float(rectform(exp(2*%pi*%i*p/q))); iLength:1000; s:GiveListOfCriticalPoints(f(z)); multiplicities; length(s); Orbits:[]; for i:1 thru length(s) step 1 do ( Orbit:GiveOrbit(s[i],iLength), Orbits:append(Orbit,Orbits) ); /*-----------------------------------------------------------------------*/ load(draw); /* ( interface to gnuplot ) by Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates */ draw2d( title = concat("Critical orbit for f(z)=z^2 + z*",string(m), " where p/q=", string(p/q)), terminal = svg, user_preamble = "set size square", /* */ file_name = concat("~/maxima/parabolic/critical_orbits/z2plusmz/1over3/arv/",string(iLength),"ss"), pic_width = 1000, /* Since Maxima 5.23, pic_width and pic_height are deprecated. */ pic_height = 1000, /* See option dimensions. To get the same effect, write dimensions=[800,600] */ /* yrange = [-0.6,0.6], xrange = [-0.6,0.6],*/ xlabel = "z.re ", ylabel = "z.im", /* vectors */ head_both = false, head_length = 0.000001, line_width = 0.3, head_angle = 1, head_type = nofilled, line_type = dots, key = "attracting vectors", color = yellow, first(va), key="", rest(va), key = "repelling vectors", color = gray, first(vr), key="", rest(vr), point_type = filled_circle, points_joined = false, point_size = 0.7, key=" critical orbit ", color =red, points(Orbits), point_size = 1.2, key= "critical points", color = blue, points(map(realpart,s),map(imagpart,s)), key= "fixed parabolic point", color = black, points([[0,0]]) );
new src code[edit]
for 5.38
kill(all); remvalue(all); /*------------- functions definitions ---------*/ /* function */ f(z):=z^2 +m*z; GiveListOfCriticalPoints(fun):= block( [d,s], /* derivative */ d:diff(fun,z,1), /* critical points z: d=0 */ s:solve(d=0,z), /* remove "z=" from list s */ s:map('rhs,s), /* convert to form x+y*%i */ s:map('rectform,s), s:map('float,s), return(s) )$ /* f(z) is used as a global function I do not know how to put it as a argument */ GiveOrbit(z0,OrbitLength):= block( [z,Orbit], z:z0, Orbit:[[realpart(z),imagpart(z)]], for i:1 thru OrbitLength step 1 do ( z:expand(f(z)), Orbit:endcons([realpart(z),imagpart(z)],Orbit)), return(Orbit) )$ /* find fixed points returns a list */ GiveFixedPoints():= block ( [s], s:solve(f(z)=z), /* remove "z=" from list s */ s:map('rhs,s), s:map('rectform,s), s:map('float,s), return(s) )$ /* riotorto.users.sourceforge.net/gnuplot/vectors/index.html */ GiveAVector(m):=block( [x,y,dx,dy], x:0, y:0, dx:realpart(m), dy:imagpart(m), vector([dx,dy],[-dx,-dy]) )$ GiveRVector(m):=block( [x,y,dx,dy], x:0, y:0, dx:realpart(m), dy:imagpart(m), vector([x,y],[dx,dy]) )$ compile(all); /* -----const values ------- */ p:1; q:3; m:1; petals:m*q; /* ??? */ /* f^q */ zq:z; for i:1 thru q step 1 do zq:f(zq); zq:expand(zq); /* Take k term in the expansion of f^q */ k:m*q+1; kCoeff:coeff(zq,z,k); /* vectors attracting and repelling http://math.stackexchange.com/questions/361205/what-is-the-shape-of-parabolic-critical-orbit */ B:petals*kCoeff*z^petals; /* http://math.stackexchange.com/questions/361205/what-is-the-shape-of-parabolic-critical-orbit */ /* attracting vectors */ va:solve(B=-1); va:map(rhs,va); va:map(rectform,va); va:map('float,va); va:map(GiveAVector,va); /* repelling vectors */ vr:solve(B=1); vr:map(rhs,vr); vr:map(rectform,vr); vr:map('float,vr); vr:map(GiveRVector,vr); /* ------------- main = computations -----------------*/ /* multiplier of fixed point = coefficient of function f */ m:float(rectform(exp(2*%pi*%i*p/q))); iLength:1000; s:GiveListOfCriticalPoints(f(z)); multiplicities; length(s); Orbits:[]; for i:1 thru length(s) step 1 do ( Orbit:GiveOrbit(s[i],iLength), Orbits:append(Orbit,Orbits) ); /*-----------------------------------------------------------------------*/ load(draw); /* ( interface to gnuplot ) by Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates */ draw2d( title = concat("Critical orbit for f(z)=z^2 + z*",string(m), " where p/q=", string(p/q)), terminal = svg, user_preamble = "set size square", /* */ file_name = concat("~/maxima/batch/criticalorbit/parabolic/1over3/",string(iLength),"ss"), dimensions = [1000,1000], /* Since Maxima 5.23, pic_width and pic_height are deprecated. */ /* yrange = [-0.6,0.6], xrange = [-0.6,0.6],*/ xlabel = "z.re ", ylabel = "z.im", /* vectors */ head_both = false, head_length = 0.000001, line_width = 0.3, head_angle = 1, head_type = filled, line_type = solid, line_width = 5, key = "attracting vectors", color = yellow, first(va), key="", rest(va), key = "repelling vectors", color = gray, first(vr), key="", rest(vr), point_type = filled_circle, points_joined = false, point_size = 0.7, key=" critical orbit ", color =red, points(Orbits), point_size = 1.2, key= "critical points", color = blue, points(map(realpart,s),map(imagpart,s)), key= "fixed parabolic point", color = black, points([[0,0]]) );
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 14:21, 21 April 2013 | 1,000 × 1,000 (110 KB) | Soul windsurfer (talk | contribs) | thicker vectors | |
14:14, 21 April 2013 | 1,000 × 1,000 (110 KB) | Soul windsurfer (talk | contribs) | {{Information |Description ={{en|1=Critical orbit for f(z)=z^2 + mz where p/q=1/3}} |Source ={{own}} |Author =Adam majewski |Date =2013-04-21 |Permission = |other_versions = }} |
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- Fractals/Iterations in the complex plane/Fatou coordinate for f(z)=z^2 + c
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- Fractals/Iterations in the complex plane/critical orbit
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