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Summary
DescriptionMiimcr.png | Julia set and critical orbit for f(z)=z*z-0.11+0.65569999i; . Made in Maxima using MIIM. Code for finding period is based on code by JC Sprott[1] |
Source | self-made, below |
Author | Adam majewski |
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Info
Orbit of critical point tends to period 3 attracting cycle :
- 0.17434140717278*%i-0.23080799291989
- 0.57522120945524*%i-0.087122596659277
- 0.55547045915754*%i-0.4332890929585
Absolute value of multiplier of above cycle is 0.94840556944351
Maxima source code
/* this is batch file for Maxima 5.13.0 http://maxima.sourceforge.net/ tested in wxMaxima 0.7.1 using draw package ( interface to gnuplot ) to draw on the screen ---------- draws Julia set for f(z)=z*z+c using modified IIM modification: stop reversed iteration if pixel was hit in reversed orbit > HitLimit times --- draws also critical orbit and finds period Adam Majewski */ c:-0.11+0.65569999*%i; /* c:-0.2-0.7*%i; */ HitLimit:10; /* proportional to number of details and time of drawing */ /* resolution is proportional to number of details and time of drawing */ iXmax:1000; iYmax:1000; /* define image size : width:iXmax-0+1; heght:iYmax-0+1 ; */ start:elapsed_run_time (); f(z,c):=z*z+c; finverseplus(z,c):=sqrt(z-c); finverseminus(z,c):=-sqrt(z-c); /* define z-plane ( dynamical ) */ zxMin:-1.6; zxMax:1.6; zyMin:-1.6; zyMax:1.6; /* */ 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 */ /* for iX:0 thru iXmax step 1 do for iY:0 thru iYmax step 1 do Hits[iX,iY]:0; */ /* compute fixed points of f(z,c) : z=f(z,c) */ fixed:float(rectform(solve([z*z+c=z],[z]))); /* Find which is repelling */ if (abs(2*rhs(fixed[1]))<1) then block( beta:rhs(fixed[1]), alfa:rhs(fixed[2]) ) else block( alfa:rhs(fixed[1]), beta:rhs(fixed[2]) ); /* ----------- find period of orbit --------------------------------------*/ z0:0; iMax:100; eps:0.01; /* first point */ z:z0; orbit:[z]; /* ------------------- compute forward orbit -----------------------------*/ for i:1 thru iMax step 1 do block ( z:f(z,c), orbit:endcons(z,orbit) /* disp(rectform(orbit[i])) */ ); IsEqual(c1,c2,eps):= if abs(realpart(c1)-realpart(c2))<=eps and abs(imagpart(c1)-imagpart(c2))<=eps then true else false; GivePeriod(orbit,eps):= block ( period:0, iLast:length(orbit), i:iLast, block ( loop, i:i-1, if not IsEqual(orbit[iLast],orbit[i],eps) then go(loop), period:iLast-i ) ); period:GivePeriod(orbit,eps); /*-------------- save orbit to draw it later on the screen ----------------------------- */ /* save the z values to 2 lists */ x0:makelist (realpart(z0), i, 1, 1); /* list of re(z) */ y0:makelist (imagpart(z0), i, 1, 1); /* list of im(z) */ for i:2 thru length(orbit) step 1 do block ( x0:cons(realpart(orbit[i]),x0), y0:cons(imagpart(orbit[i]),y0) ); NumberOfPoints:0; /* choose repeller as a starting point */ /*save beta in stack */ stack:[beta]; /* make 2 list of points and copy beta to to lists */ xx:makelist (realpart(beta), i, 1, 1); /* list of re(z) */ yy:makelist (imagpart(beta), i, 1, 1); /* list of im(z) */ NumberOfPoints:1; /* reversed iteration of beta */ block ( 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<HitLimit then block( Hits[iX,iY]:hit+1, stack:endcons(z,stack), /* push = add z at the end of list stack */ if hit=0 then block( xx:cons(realpart(z),xx), yy:cons(imagpart(z),yy)), NumberOfPoints:NumberOfPoints+1 ), /*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<HitLimit then block( 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 block( xx:cons(realpart(z),xx), yy:cons(imagpart(z),yy)), /* draw point */ NumberOfPoints:NumberOfPoints+1 ), if is(not emptyp(stack)) then go(loop) ); stop:elapsed_run_time (); t:fix(stop-start); /* draw reversed orbit of beta using draw package */ load(draw); draw2d( file_name = "miimcr_", terminal = 'png, pic_width = iXmax, pic_height = iYmax, yrange = [zyMin,zyMax], xrange = [zxMin,zyMax], title= "Julia set in dynamical plane for f(z,c):=z*z+c drawn using MIIM", label ([concat("c=",string(c),"; Period=",string(period),". Resoution=",string(iXmax), " Time=",string(t),"HitLimit=",string(HitLimit)),0.0,-1.5]), xlabel = "Z.re ", ylabel = "Z.im", point_type = filled_circle, point_size = 0.3, color = black, key = concat(string(NumberOfPoints)," points of Julia set."), points(xx,yy), color =red, point_size = 0.3, key = concat("Points of critical orbit"), points(x0,y0) );
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current | 19:43, 5 March 2008 | ![]() | 1,000 × 1,000 (18 KB) | Soul windsurfer | {{Information |Description= |Source=self-made |Date= |Author= Adam majewski |Permission= |other_versions= }} |
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