File:Multiplier4 f.png
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Summary
DescriptionMultiplier4 f.png |
English: Stability index of periodic z-points for c along horizontal axis for fc(z)= z*z +c
Polski: Indeks stabilności dla parametrów c wzdłuż osi OX dla fc(z) = z*z +c |
Date | |
Source | Own work |
Author | Adam majewski |
Summary
This image shows some features of the discrete dynamical system
based on complex quadratic polynomial :
- .
When coeeficient goes inside Mandelbrot set from c=0.25 to c=-2 along horizontal axis ( imaginary part of c is zero) then stability index of periodic points is changing. This diagram shows what happens there.
- is zero for centers of hyperboic components of Mandelbrot set, like c= 0 or c = -1
- is 1 for root points, like c=0.25, c=-0.75 or c=-1.25
Maxima src code
load(cpoly); /* for bfallroots */ /* ################ Definitions ############################ */ /* use : load(cpoly); roots:GiveRoots_bf(eq); */ GiveRoots_bf(g):= block( [cc:bfallroots(expand(g)=0)], cc:map(rhs,cc),/* remove string "c=" */ return(cc) )$ /* functions for computing stability index of fixed points ; returns real value */ stability_beta(_c):= block( beta:(1+sqrt(abs(1-4*_c)))/2, stability:abs(float(2*beta)), /* abs(multiplier(z)) */ return(stability) ); stability_alfa(_c):= block( alfa:(1-sqrt(abs(1-4*_c)))/2, stability:abs(float(2*alfa)), /* abs(multiplier(z)) */ return(stability) ); stability_2(c):= block( [eq,roots,z2], eq:z*z +z +c +1, roots:GiveRoots_bf(eq), z2:roots[1], m:4*z2*(z2*z2+c), /* multiplier */ stability:cabs(float(m)), /* abs(multiplier(z2)) */ return(stability) ); stability_3(c):= block( [eq,roots,z3], eq:z^6+z^5+(3*c+1)*z^4+(2*c+1)*z^3+(3*c^2+3*c+1)*z^2+(c^2+2*c+1)*z+c^3+2*c^2+c+1, roots:GiveRoots_bf(eq), z3:roots[1], m:8*z3*(z3^2+c)*((z3^2+c)^2+c), /* multiplier */ stability:cabs(float(m)), /* abs(multiplier(z3)) */ return(stability) ); stability_4(c):= block( [eq,roots,z4], eq:z^12+6*c*z^10+z^9+(15*c^2+3*c)*z^8+4*c*z^7+(20*c^3+12*c^2+1)*z^6+(6*c^2+2*c)*z^5+ (15*c^4+18*c^3+3*c^2+4*c)*z^4+(4*c^3+4*c^2+1)*z^3+ (6*c^5+12*c^4+6*c^3+5*c^2+c)*z^2+(c^4+2*c^3+c^2+2*c)*z+c^6+3*c^5+3*c^4+3*c^3+2*c^2+1, roots:GiveRoots_bf(eq), z4:roots[1], m:16*z4*(z4^2+c)*((z4^2+c)^2+c)*(((z4^2+c)^2+c)^2+c), /* multiplier */ stability:cabs(float(m)), /* abs(multiplier(z3)) */ return(stability) ); /* root points */ root_pts:[-1.401154502237,-1.401151982029,-1.401140214699,-1.4010852713,-1.4008287424,-1.3996312389, -1.3940461566,-1.3680989394,-1.25,-0.75]; ry:[1,1,1,1,1,1,1,1,1,1]; /* centers of hyperbolic components of F1/2 family ( on real axis) */ centers:[-1.381547484432061460,-1.31070264133683,-1,0]; /* periods: 8,4,2,1 */ cy:[0,0,0,0]; xMax:0.25; xMin:-1.39; yMin:0; yMax:xMax-xMin; /* for square image, but yMax can be > (xMax-xMin) */ /* ------------------------------- main ------------------------------------------------- */ load(draw); draw2d( terminal = png, file_name = "multiplier4_f", pic_height= 1100, pic_width=1100, user_preamble="set bmargin 7;set rmargin 25;set key out;set key top right", title = "Stability index of periodic z-points for c along horizontal axis for fc(z)= z*z +c ", ylabel = "Stability index of z", xlabel = "c", yrange = [yMin,yMax], /* period 1 */ key = "period 1 beta", color = red, explicit(stability_beta,x,xMin,xMax), key = "period 1 alfa", color = blue, explicit(stability_alfa,x,xMin,xMax), /* period 2 */ key = "period 2 ", color = green, explicit(stability_2,x,xMin,xMax), /* period 3 */ key = "period 3 ", color = magenta, explicit(stability_3,x,xMin,xMax), /* period 4 */ key = "period 4 ", color = purple, explicit(stability_4,x,xMin,xMax), /* grid and tics */ xtics = {-2,-1.94079980652948,-1.75487766624669,-1.3680989394,-1.31070264133683,-1.25,-1,-3/4,0,0.25}, /* -2,root points,centers, 0 */ xtics_axis = true, /* plot tics on x-axis */ xtics_rotate = true, ytics = {0,1}, grid = true, /* draw grid*/ /* special points */ point_type = filled_circle, point_size = 0.5, points_joined = false, color = black, key = "root points", points(root_pts,ry), color = red, key = "centers", points(centers,cy) )$
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20 June 2009
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 10:12, 20 June 2009 | 1,100 × 1,100 (17 KB) | wikimediacommons>Soul windsurfer | {{Information |Description={{en|1=Stability index of periodic z-points for c along horizontal axis for fc(z)= z*z +c}} {{pl|1=Indeks stabilności dla parametrów c wzdłuż osi OX dla fc(z) = z*z +c}} |Source=Own work by uploader |Author=[[User:Adam maje |
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