Summary

Summary

This image shows some features of the discrete dynamical system

${\displaystyle z_{n+1}=f_{c}(z_{n})\,}$

based on complex quadratic polynomial :

${\displaystyle f_{c}(z)=z^{2}+c.\,}$.

When coeeficient ${\displaystyle c\,}$ 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.

Stability index:

• 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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