File:Critical orbits for f(z)=z^4-iz.png
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![File:Critical orbits for f(z)=z^4-iz.png](https://upload.wikimedia.org/wikipedia/commons/thumb/1/18/Critical_orbits_for_f%28z%29%3Dz%5E4-iz.png/600px-Critical_orbits_for_f%28z%29%3Dz%5E4-iz.png?20130315203138)
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[edit]DescriptionCritical orbits for f(z)=z^4-iz.png |
English: Critical orbits for f(z)=z^4-iz . Because critical orbit tends to fixed point very slowly there is a gap between critical orbit and fixed point. |
Date | |
Source | Own work |
Author | Adam majewski |
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[edit]Maxima CAS src code
[edit]kill(all); remvalue(all); /*------------- functions definitions ---------*/ /* function */ f(z):=z^4+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) )$ compile(all); /* ------------ */ fs:string(f(z)); p:3; q:4; m:rectform(exp(2*%pi*%i*p/q)); iLength:100000; 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 */ /* http://riotorto.users.sourceforge.net/gnuplot/ps/index.html */ draw2d( title = concat("All critical orbits for discrete map f(z)=",fs ," where m=e^{2*pi*i*3/4}= -i "), terminal = png, user_preamble = "set angles degrees; set grid polar 15; set xtics 0.62996052494744; set mxtics 0.5; set size square", /* 360/12-30 */ file_name = concat("~/maxima/parabolic/critical_orbits/z4plusmz/3over4/",string(iLength),"b7"), 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 = [-0.75,0.75], xrange = [-0.75,0.75], xlabel = "z.re ", ylabel = "z.im", 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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current | 20:31, 15 March 2013 | ![]() | 1,000 × 1,000 (35 KB) | Soul windsurfer (talk | contribs) | more points |
10:22, 10 March 2013 | ![]() | 1,000 × 1,000 (34 KB) | Soul windsurfer (talk | contribs) | User created page with UploadWizard |
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