This diagram shows function (red 3D curve):

${\displaystyle \mathrm {csqrt} :\mathbb {R} \to \mathbb {C} \,}$

Gray 2D lines are projections of 3D red curve on 2D planes.

One can see that red curve consist of 2 parabolas rotated with respect to themselves ( 90 degrees)

The image was made using

• Maxima CAS program
• draw package archive copy at the Wayback Machine by Mario Rodriguez Riotorto ( interface to Gnuplot)
• Gnuplot

Tested on versions:

• wxMaxima 0.7.6
• Maxima 5.16.3
• Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
• Gnuplot Version 4.2 patchlevel 3

After running below batch file image was created on the screen. It was manually rotated, copied to clipboard, pasted to Paint program. Some interrupted gray lines were added.

Is it a explanation of period doubling ?

"It is an explanation of the shape of the curves in period doubling, but there is more behind period doubling. E.g., when the multiplier of an n-periodic point z is -1, then the equation for period 2*n has a multiple solution. " Wolf Jung

 /* functions for computing  */ 
 give_beta(_c):= sqrt(_c) $  

 give_alfa(_c):= -sqrt(_c) $ 

 /* c */
 xMax:1;
 xMin:-1;

 /* re(s) */
 yMin:-2;
 yMax:2;

 /* im(s) */
 zMin:-3;
 zMax:3; 
 iXmax:1000;
 dx:(xMax-xMin)/iXmax;
 p1_alfa:[];
 p1_beta:[];
 re_c:[];
 im_c:[];
 re_im:[];

 for c:xMin step dx thru xMax do 
 (
 alfa:give_alfa(c),
 p1_alfa:cons([c,realpart(alfa),imagpart(alfa)],p1_alfa),
 re_c:cons([c,realpart(alfa),zMin],re_c),
 im_c:cons([c,-2,imagpart(alfa)],im_c),
 re_im:cons([-1,realpart(alfa),imagpart(alfa)],re_im),
 beta:give_beta(c),
 p1_beta:cons([c,realpart(beta),imagpart(beta)],p1_beta),
 re_c:cons([c,realpart(beta),zMin],re_c),
 im_c:cons([c,-2,imagpart(beta)],im_c),
 re_im:cons([-1,realpart(beta),imagpart(beta)],re_im)
 ); 

 load(draw);
 draw3d(
  user_preamble="set ticslevel 0;",
  terminal = screen,
  pic_height= iXmax,
  title       = "square roots s= sqrt(c) of real number c ",
  ylabel     = "Re(sqrt(c))",
  zlabel ="Im(sqrt(c))",
  xlabel     = "c",
  yrange = [yMin,yMax],
  zrange = [zMin,zMax],
  point_type    = filled_circle,
  /* */
  points_joined = false,
  color=gray,
  point_size    = 0.1,
  points(re_c),
  points(im_c),
  points(re_im),
  /* */
  point_size    = 0.2,
  points_joined = true,
  color  = red,
  points(p1_beta),
  points(p1_alfa),
  /* grid and tics */
  xtics      = {-2,-1,0,1,2}, 
  xtics_rotate = true,
  ytics      = {-2,-1,0,1,2},
  ztics		={-2,-1,0,1,2},
  grid       = true /* draw grid*/
 )$

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