File:Other domains for Fatou coordinate for f(z) = z+a 2z ^2+O(z^3).svg
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Summary[edit]
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 |
Licensing[edit]
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
compare with[edit]
M 597 LECTURE NOTES TOPICS IN MATHEMATICS COMPLEX DYNAMICS by LUKAS GEYER, page 39
Maxima CAS src code[edit]
/* 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[edit]
- Replace
rgb\( 0, 0, 0\)
with#000
to prevent black dots from being removed in the next step. - Remove
\t?<u[^>]*\)'/>\n
.
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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 (talk | contribs) | 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 (talk | contribs) | 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 |