File:Paritition of dynamic plane of quadratic polynomial for 1 6.svg
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Summary
DescriptionParitition of dynamic plane of quadratic polynomial for 1 6.svg |
English: Paritition of dynamic plane of critically preperiodic quadratic polynomial for external ray t = 1/6 landing at the critical value z= c = i. Preperiod = 1 period = 2 |
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Source | Own work |
Author | Adam majewski |
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Maxima CAS src code
/* batch file for Maxima CAS maxima batch("r.mac") ----------- Wolf Jung program Mandel ==================== The angle 1/6 or 0p01 has preperiod = 1 and period = 2. The corresponding parameter ray is landing at a Misiurewicz point of preperiod 1 and period dividing 2. Do you want to draw the ray and to shift c to the landing point? */ start:elapsed_run_time (); kill(all); remvalue(all); /* --------------------------definitions of functions ------------------------------*/ f(z,c):=z*z+c; /* Complex quadratic map */ finverseplus(z,c):=sqrt(z-c)$ finverseminus(z,c):=-sqrt(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)$ /*Standard polynomial F_p \, which roots are periodic z-points of period p and its divisors */ F(p, z, c) := fn(p, z, c) - z $ /* Function for computing reduced polynomial G_p\, which roots are periodic z-points of period p without its divisors*/ G[p,z,c]:= block( [f:divisors(p), t:1], /* t is temporary variable = product of Gn for (divisors of p) other than p */ f:delete(p,f), /* delete p from list of divisors */ if p=1 then return(F(p,z,c)), for i in f do t:t*G[i,z,c], g: F(p,z,c)/t, return(ratsimp(g)) )$ GiveRoots(g):= block( [cc], cc:bfallroots(expand(%i*g)=0), cc:map(rhs,cc),/* remove string "c=" */ cc:map('float,cc), return(cc) )$ /* endcons the complex point to list in the format for draw package */ endconsD(point,list):=endcons([realpart(point),imagpart(point)],list)$ consD(point,list):=cons([realpart(point),imagpart(point)],list)$ GiveForwardOrbit(z0,c,iMax):= /* computes (without escape test) forward orbit of point z0 and saves it to the list for draw package */ block( [z,orbit,temp], z:z0, /* first point = critical point z:0+0*%i */ orbit:[[realpart(z),imagpart(z)]], for i:1 thru iMax step 1 do ( z:expand(f(z,c)), orbit:endcons([realpart(z),imagpart(z)],orbit)), return(orbit) )$ /* Gives points of backward orbit of z=repellor */ GiveBackwardOrbit(c,repellor,zxMin,zxMax,zyMin,zyMax,iXmax,iYmax):= block( hit_limit:25, /* proportional to number of details and time of drawing */ PixelWidth:(zxMax-zxMin)/iXmax, PixelHeight:(zyMax-zyMin)/iYmax, /* 2D array of hits pixels . Hit > 0 means that point was in orbit */ array(Hits,fixnum,iXmax,iYmax), /* no hits for beginning */ /* choose repeller z=repellor as a starting point */ stack:[repellor], /*save repellor in stack */ /* save first point to list of pixels */ x_y:[repellor], /* reversed iteration of repellor */ loop, /* pop = take one point from the stack */ z:last(stack), stack:delete(z,stack), /*inverse iteration - first preimage (root) */ z:finverseplus(z,c), /* translate from world to screen coordinate */ iX:fix((realpart(z)-zxMin)/PixelWidth), iY:fix((imagpart(z)-zyMin)/PixelHeight), hit:Hits[iX,iY], if hit<hit_limit then ( Hits[iX,iY]:hit+1, stack:endcons(z,stack), /* push = add z at the end of list stack */ if hit=0 then x_y:endcons( z,x_y) ), /*inverse iteration - second preimage (root) */ z:-z, /* translate from world to screen coordinate, coversion to integer */ iX:fix((realpart(z)-zxMin)/PixelWidth), iY:fix((imagpart(z)-zyMin)/PixelHeight), hit:Hits[iX,iY], if hit<hit_limit then ( Hits[iX,iY]:hit+1, stack:endcons(z,stack), /* push = add z at the end of list stack to continue iteration */ if hit=0 then x_y:endcons( z,x_y) ), if is(not emptyp(stack)) then go(loop), return(x_y) /* list of pixels in the form [z1,z2] */ )$ /*-----------------------------------*/ Psi_n(r,t,z_last, Max_R):= /* */ block( [iMax:200, iMax2:0], /* ----- forward iteration of 2 points : z_last and w --------------*/ array(forward,iMax-1), /* forward orbit of z_last for comparison */ forward[0]:z_last, i:0, while cabs(forward[i])<Max_R and i< ( iMax-2) do ( /* forward iteration of z in fc plane & save it to forward array */ forward[i+1]:forward[i]*forward[i] + c, /* z*z+c */ /* forward iteration of w in f0 plane : w(n+1):=wn^2 */ r:r*2, /* square radius = R^2=2^(2*r) because R=2^r */ t:mod(2*t,1), /* */ iMax2:iMax2+1, i:i+1 ), /* compute last w point ; it is equal to z-point */ R:2^r, /* w:R*exp(2*%pi*%i*t), z:w, */ array(backward,iMax-1), backward[iMax2]:rectform(ev(R*exp(2*%pi*%i*t))), /* use last w as a starting point for backward iteration to new z */ /* ----- backward iteration point z=w in fc plane --------------*/ for i:iMax2 step -1 thru 1 do ( temp:float(rectform(sqrt(backward[i]-c))), /* sqrt(z-c) */ scalar_product:realpart(temp)*realpart(forward[i-1])+imagpart(temp)*imagpart(forward[i-1]), if (0>scalar_product) then temp:-temp, /* choose preimage */ backward[i-1]:temp ), return(backward[0]) )$ GiveRay(t,c):= block( [r], /* range for drawing R=2^r ; as r tends to 0 R tends to 1 */ rMin:1E-10, /* 1E-4; rMin > 0 ; if rMin=0 then program has infinity loop !!!!! */ rMax:2, caution:0.9330329915368074, /* r:r*caution ; it gives smaller r */ /* upper limit for iteration */ R_max:300, /* */ zz:[], /* array for z points of ray in fc plane */ /* some w-points of external ray in f0 plane */ r:rMax, while 2^r<R_max do r:2*r, /* find point w on ray near infinity (R>=R_max) in f0 plane */ R:2^r, w:rectform(ev(R*exp(2*%pi*%i*t))), z:w, /* near infinity z=w */ zz:cons(z,zz), unless r<rMin do ( /* new smaller R */ r:r*caution, R:2^r, /* */ w:rectform(ev(R*exp(2*%pi*%i*t))), /* */ last_z:z, z:Psi_n(r,t,last_z,R_max), /* z=Psi_n(w) */ zz:cons(z,zz) ), return(zz) )$ /* converts complex number z = x*y*%i to the list in a draw format: [x,y] */ d(z):=[float(realpart(z)), float(imagpart(z))]$ ToPoints(myList):= points(map(d,myList))$ /* give Draw List from one point*/ ToPoint(z):=points([d(z)])$ compile(all)$ /* ----------------------- main ----------------------------------------------------*/ /* external angle in turns */ /* resolution is proportional to number of details and time of drawing */ iX_max:1000$ iY_max:1000$ /* define z-plane ( dynamical ) */ ZxMin:-2.0$ ZxMax:2.0$ ZyMin:-2.0$ ZyMax:2.0$ /* give c a value */ c:%i$ /* */ t:1/6; /* compute fixed points */ Beta:float(rectform((1+sqrt(1-4*c))/2))$ /* compute repelling fixed point beta */ alfa:float(rectform((1-sqrt(1-4*c))/2))$ /* other fixed point */ /* compute backward orbit of repelling fixed point */ xy: GiveBackwardOrbit(c,Beta,ZxMin,ZxMax,ZyMin,ZyMax,iX_max,iY_max)$ /* compute ray points & save to zz list */ eRay : GiveRay(t,c)$ eRayT:GiveRay(t/2,c)$ eRayTp:GiveRay((t+1)/2,c)$ /* time of computations */ time:fix(elapsed_run_time ()-start)$ /* draw it using draw package by */ load(draw)$ path:"~/maxima/batch/julia/knead/"$ /* if empty then file is in a home dir */ /* if graphic file is empty (= 0 bytes) then run draw2d command again */ draw2d( terminal = 'svg, file_name = sconcat(path,"k25_"), user_preamble="set size square;set key top right", title= concat("Dynamical plane for fc(z)=z*z+",string(c)), dimensions = [iX_max, iY_max], yrange = [ZyMin,ZyMax], xrange = [ZxMin,ZyMax], xlabel = "Z.re ", ylabel = "Z.im", point_type = filled_circle, points_joined =true, point_size = 0.2, color = red, points_joined =false, color = black, key = "backward orbit of z=beta", points(map(realpart,xy),map(imagpart,xy)), points_joined =false, color = green, point_size = 1.4, key = "critical value", ToPoint(c), key = sconcat("external ray t=",string(t)), color = green, points_joined =true, point_size = 0.2, ToPoints(eRay), points_joined = false, color = black, point_size = 1.4, key = "critical point z = 0.0", ToPoint(0.0), points_joined =true, point_size = 0.2, color = red, key = sconcat("external ray t/2 = ", string(t/2)), ToPoints(eRayT), key = sconcat("external ray (t+1)/2 =",string((t+1)/2)), color = magenta, ToPoints(eRayTp) )$
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13 July 2019
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 20:17, 13 July 2019 | 1,000 × 1,000 (725 KB) | Soul windsurfer | User created page with UploadWizard |
File usage
The following 6 pages use this file:
- Fractals/Iterations in the complex plane/MandelbrotSetExterior/ParameterExternalRay
- Fractals/Iterations in the complex plane/def cqp
- Fractals/Iterations in the complex plane/dynamic external rays
- Fractals/Iterations in the complex plane/jlamination
- Fractals/Iterations in the complex plane/misiurewicz
- Fractals/mandel
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Short title | Gnuplot |
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Image title | Produced by GNUPLOT 5.3 patchlevel 0 |
Width | 1000 |
Height | 1000 |