File:Other domains for Fatou coordinate for f(z) = z+a 2z ^2+O(z^3).svg
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DescriptionOther domains for Fatou coordinate for f(z) = z+a 2z ^2+O(z^3).svg |
English: Other domains for Fatou coordinate for f(z) = z+a 2z ^2+O(z^3) |
Date | |
Source | Own work |
Author | Adam majewski |
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I, the copyright holder of this work, hereby publish it under the following license:
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compare with
M 597 LECTURE NOTES TOPICS IN MATHEMATICS COMPLEX DYNAMICS by LUKAS GEYER, page 39
Maxima CAS src code
/* b batch file for maxima There are 2 complex planes : * u-plane * z-plane 1 step transformation from u plane go to z plane using z=h(u) */ kill(all); remvalue(all); /* */ iMax:1000; /* number of points to draw */ uxMin: -80; /* */ /* =================== functions ============ */ /* complex quadratic polynomial with fixed point z=0 and unique critical point z=-1/2 */ p(z):=z+z*z$ /* uy:f(ux) */ f(ux):= (a*ux + b ); /* only line segment */ /* no then else sqrt(r2-ux*ux); circle segment */ /* 1 step transformation from u plane go to z plane using z=h(u) h(0) expt: undefined: 0 to a negative exponent. -- an error. To debug this try: debugmode(true); h(u):= if (u=0.0) then infinity else -1/u$ */ h(u):= if (u=0.0) then infinity else -1/u$ m(u):= -realpart(u)+imagpart(u)*%i$ /* minus */ /* inverse function of f ux = fi(uy) */ fi(uy):=block ( [s,ux], if (uy>uyt) then s: (uy - b)/a else s:sqrt(r2-uy*uy), s:float(s), return(s) )$ /* converts complex number into list for draw package */ draw_format(z):= if (z=infinity) then [1000.0,1000.0] else [float(realpart(z)),float(imagpart(z))]; /* line im(u) = const in a draw format : point(list ) uu is a list of u values zz is a list of z values u = ux + uy*%i */ GiveHorizontalLines(uy):= block( [uu,zz], uu:makelist ( uy*%i + (1.0 +k/10), k, -1000, 1000 ), zz:map(h,uu), uu:map(draw_format,uu), zz:map(draw_format,zz), [points(uu),points(zz)] /* list of 2 sublists : first is a uuh list , second is a zzh list */ )$ compile(all); /* ============== compute =============== */ zc :-1/2; /* critical point */ a:-1; b:2; uxMax: b; uxStep: (uxMax-uxMin)/iMax; /* point to point method of drawing compute first point of curve, create list and save point to this list */ /* A1 */ uListA1:[]; for ux:uxMin step uxStep while (ux<= uxMax) do ( uy:f(ux), uListA1:cons(ux+uy*%i,uListA1), uListA1:endcons(ux-uy*%i,uListA1) )$ uListR1: map(m, uListA1)$ zListA1:map(h,uListA1)$ zListR1: map(m, zListA1)$ /* A2 */ b:b+2; uxMax: b; uxStep: (uxMax-uxMin)/iMax; uListA2:[]; for ux:uxMin step uxStep while (ux<= uxMax) do ( uy:f(ux), uListA2:cons(ux+uy*%i,uListA2), uListA2:endcons(ux-uy*%i,uListA2) )$ zListA2: map(h,uListA2)$ /* A3 */ b:b+2; uxMax: b; uxStep: (uxMax-uxMin)/iMax; uListA3:[]; for ux:uxMin step uxStep while (ux<= uxMax) do ( uy:f(ux), uListA3:cons(ux+uy*%i,uListA3), uListA3:endcons(ux-uy*%i,uListA3) )$ zListA3 : map(h,uListA3)$ /* horizontal lines without u=0 */ uh:[1,2,3,4,5,6,-1,-2,-3,-4,-5,-6]$ /* list of values for lines im(u) = const */ uzh:map(GiveHorizontalLines,uh)$ uuh:map(first,uzh)$ zzh:map(second,uzh)$ /* critical orbit */ critical:[]$ z: zc; for i:1 thru 20 step 1 do ( critical:cons(z,critical), z:p(z) )$ /* single important points */ z0 : 0; /* origin z=0 */ /* convert list to draw format */ uListA3:map(draw_format,uListA3)$ uListA2:map(draw_format,uListA2)$ uListA1:map(draw_format,uListA1)$ uListR1:map(draw_format,uListR1)$ zListA3:map(draw_format,zListA3)$ zListA2:map(draw_format,zListA2)$ zListA1:map(draw_format,zListA1)$ zListR1:map(draw_format,zListR1)$ critical : map ( draw_format, critical)$ z0:draw_format(z0); /* ================= draw ======================================*/ path:"~/maxima/batch/julia/parabolic/1over1/petal3/"$ /* pwd ; if you put here working directory name then graphic file will be saved in that dir */ FileName:concat(string(b),"6b")$ /* without extension which is the terminal name */ load(draw); /* Mario Rodríguez Riotorto http://riotorto.users.sourceforge.net/gnuplot/points/index.html */ draw( terminal = 'svg, file_name = concat(path,FileName), columns = 2, dimensions=[1000,500], /* x = y*columns */ gr2d(title = " u plane ", /* xrange = [0,3], xtics={-10, -2, 0,2}, */ yrange = [-20.0,20.0], xrange = [-20.0,20.0], grid = false, xaxis = false, points_joined =true, point_size = 0.2, point_type = filled_circle, color=green, points(uListA2), color = yellow, points(uListA3), color = red, points(uListA1), color = blue, points(uListR1), /* key = "orbits = invariant cirves",*/ color = gray, uuh ), gr2d(title = " z plane : z = -1/u with petals ", yrange = [-1.2,1.2], xrange = [-1.2,1.2], points_joined =true, grid = false, point_size = 0.1, point_type = filled_circle, key = "", color = yellow, points(zListA3), key = "", color=green, points(zListA2), key = "attracting petal", color = red, points(zListA1), key = "repelling petal", color = blue, points(zListR1), key = "", /*key = "orbits = invariant cirves",*/ color = gray, zzh, points_joined =false, color = black, key="", points(critical), point_size = 0.8, key="fixed point", points([z0]) ) );
Postprocessing
- Replace
rgb\( 0, 0, 0\)
with#000
to prevent black dots from being removed in the next step. - Remove
\t?<u[^>]*\)'/>\n
.
Items portrayed in this file
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11 August 2016
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 07:30, 20 May 2018 | 1,000 × 500 (213 KB) | TilmannR | replaced <code>rgb\( 0, 0, 0\)</code> with <code>#000</code> and removed <code>\t?<u[^>]*\)'/>\n</code> | |
19:01, 11 August 2016 | 1,000 × 500 (3.98 MB) | Soul windsurfer | User created page with UploadWizard |
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Short title | Gnuplot |
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Image title | Produced by GNUPLOT 4.6 patchlevel 4 |
Width | 1000 |
Height | 500 |