File:Parabolic critical orbit of rational function ( Blaschke fraction).png
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Parabolic_critical_orbit_of_rational_function_(_Blaschke_fraction).png (600 × 600 pixels, file size: 19 KB, MIME type: image/png)
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Summary
DescriptionParabolic critical orbit of rational function ( Blaschke fraction).png |
English: Parabolic critical orbit of rational function ( Blaschke fraction) f(z) = rho * z^2 * (z-3)/(1-3*z) where rho = -0.6170144002709304 +0.7869518599370003*%i ( cusp ) |
Date | |
Source | Own work |
Author | Adam majewski |
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Maxima CAS src code
/* [0.7121885831301268*%i-0.7019881922504848, 0.6038629905954274*%i+0.7970881310050664, (-0.8738242051822692*%i)-0.4862419751909308, 0.7121885831301268*%i-0.701988192250485] */ kill(all); display2d:false; ratprint : false; /* remove "rat :replaced " */ /* 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:[z], for i:1 thru OrbitLength step 1 do ( z:expand(f(z)), if cabs(z) > 2 then ( print(z0), print("orbit of is escaping"), return(Orbit)), Orbit:endcons(z ,Orbit) ), print(z0), print("orbit is not escaping"), return(Orbit) )$ /* f(z) is used as a global function I do not know how to put it as a argument */ GiveOneArmOrbit(z0,OrbitLength):= block( [z,Orbit], z:z0, Orbit:[z], for i:1 thru OrbitLength step 1 do ( z:expand(f(z)), if cabs(z) > 2 then ( print(z0), print("orbit of is escaping"), return(Orbit)), Orbit:endcons(z ,Orbit) ), print(z0), print("orbit is not escaping"), return(Orbit) )$ /* converts angle in radians in range -Pi to Pi to turns */ GiveTurn(a):= ( if a<0 then a:a+2*%pi, /* from range (-Pi,Pi) to range (0,2Pi) */ float(a/(2*%pi)) /* from radians to turns */ )$ /* give turn of complex number z */ cturn(z):=GiveTurn(carg(z))$ /* give Draw List from one point*/ /* converts complex number z = x*y*%i to the list in a draw format: [x,y] */ d(z):=[float(realpart(z)), float(imagpart(z))]$ ToPoint(z):= points([d(z)])$ /* give Draw List from one point*/ ToPoints(myList):= points(map(d,myList))$ compile(all); radius : 1.0; t:1/3; rho : -0.6170144002709304 +0.7869518599370003*%i; f(z):= float(rectform((rho * z^2 * (z-3)/(1-3*z))))$ dz: diff(f(z),z,1); iLength:10000; Orbit: GiveOneArmOrbit(1.0,iLength)$ /* period 3 points */ e: f(f(f(z))) = z$ load (to_poly_solve); s: to_poly_solve (e,z); s: args(s); /* https://stackoverflow.com/questions/12834709/create-a-union-into-a-list-in-maxima */ s:flatten(s); s:map(rhs,s); r:[]; for z in s do if (abs(abs(z) -1) < 0.1) then r:cons(z,r); cycle1:[]; z:r[1]; cycle1: cons(z, cycle1); for i:1 thru 3 step 1 do ( z:float(rectform(f(z))), cycle1:cons(z, cycle1) ); turns:map(cturn,cycle1); Orbit : ToPoints(Orbit)$ r:ToPoints(r)$ z32: ToPoint(cycle1[2]); z31: ToPoint(cycle1[1]); z33: ToPoint(cycle1[3]); zcr: ToPoint(1); cycle1: ToPoints(cycle1)$ load(draw); /* ( interface to gnuplot ) by Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates */ draw2d( title = "Parabolic period 3 orbit for f(z)= rho * z^2 * (z-3)/(1-3*z))", terminal = png, user_preamble = "set size square; set key left top;", /* 360/26=13.85 ; 360/(2*26)=6,923 */ file_name = concat("~/Dokumenty/julia_blaszke/period3/maxima/cycle_check/1over3/", concat("cycles_j_",string(iLength))), dimensions = [600, 600], /* Since Maxima 5.23, pic_width and pic_height are deprecated. */ yrange = [-1.2,1.2], xrange = [-1.2, 1.2], xlabel = "z.re ", ylabel = "z.im", line_width = 1, nticks = 50, color = gray, transparent = true, ellipse(0,0,1,1,0,360), /* unit circle */ point_type = filled_circle, points_joined = true, point_size = 1.2, key="periodic points", color = red , cycle1, points_joined = false, point_size = 1.2, key="critical point", color = blue , zcr, point_size = 0.8, points_joined = false, key="critical orbit ", color = black , Orbit );
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20 June 2022
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current | 16:24, 20 June 2022 | 600 × 600 (19 KB) | Soul windsurfer | Uploaded own work with UploadWizard |
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Vertical resolution | 37.8 dpc |