File:Distance between point of critical orbit and alfa fixed point in parabolic cases.png
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Distance_between_point_of_critical_orbit_and_alfa_fixed_point_in_parabolic_cases.png (640 × 480 pixels, file size: 30 KB, MIME type: image/png)
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Summary
DescriptionDistance between point of critical orbit and alfa fixed point in parabolic cases.png |
English: Distance between point of critical orbit and alfa fixed point in parabolic cases |
Date | |
Source | Own work |
Author | Adam majewski |
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Maxima CAS src code
kill(all); /* ---------- functions ---------------------- */ /* http://en.wikipedia.org/wiki/Complex_quadratic_polynomial */ f(z,c):=rectform(z*z+c); fn(p, z, c) := if p=0 then z elseif p=1 then f(z,c) else f(fn(p-1, z, c),c); /* find fixed point alfa of function f(z,c) */ GiveFixed(c):= float(rectform((1-sqrt(1-4*c))/2))$ /* gives distance between 2 complex points on plane */ GivePlaneDistance(z1,z2):=float(abs(z2-z1))$ /* 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 */ ToCircle(t,r):=r*%e^(%i*t*2*%pi); /* point on boundary of period 1 component of Mandelbrot set */ GiveC(t,r):= block( [w], w:ToCircle(t,r), return(float(rectform(F(w)))) )$ GiveDrawList(Denominator, iMax):= block ([temp], /* temp:[], */ temp:GiveList(Denominator, iMax), return(temp) ); GiveList(Denominator, iMax):= block ( [zprev,znext, c, zAlfa, dist,d_list], c: GiveC(1/Denominator,1), zAlfa:GiveFixed(c), zprev:0.0, znext:fn(Denominator,zprev,c), dist:GivePlaneDistance(zAlfa,znext), d_list:[[1,dist]], zprev : znext, for i:2 step 1 thru iMax do ( znext:fn(Denominator, zprev,c), dist:GivePlaneDistance(zAlfa,znext), d_list:endcons([i,dist],d_list), zprev : znext ), return(d_list) )$ compile(all)$ /* ---------- constant ---------------------------*/ DenominatorMax:10$ iMax:5000$ /* ------------- main -----------------------*/ DenominatorsList : makelist(i,i,1,DenominatorMax)$ ColorList:[red,blue,green, black, yellow , purple , red,brown, cyan, violet ]$ DrawLists:[]$ /* empty list */ for Denominator in DenominatorsList do ( disp(Denominator), DrawList:GiveDrawList(Denominator, iMax), DrawLists:cons(points(DrawList),DrawLists), DrawLists:cons(key =concat("period ", string(Denominator)),DrawLists), DrawLists:cons(color=ColorList[Denominator],DrawLists) )$ /* ----------------- draw ------------ */ load(draw); draw2d( terminal = png, file_name = "dist", title = "Distance between point of critical orbit and alfa fixed point in parabolic cases ", user_preamble = "", xlabel = "rotation = iteration/period", ylabel = "distance from zn to alfa fixed point", points_joined =false, point_size = 0.6, point_type = filled_circle, DrawLists );
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30 March 2012
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 14:18, 30 March 2012 | 640 × 480 (30 KB) | Soul windsurfer |
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