File:Miim.jpg

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

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/*  
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
Adam Majewski
*/
c:-0.11+0.65569999*%i;   /*    */
HitLimit:14; /* proportional to number of details and time of drawing */
/* resolution is proportional to number of details and time of drawing */
iXmax:500;
iYmax:500;
/* define image size : width:iXmax-0+1;      heght:iYmax-0+1 ;    */
start:elapsed_run_time ();
NumberOfPoints:0;
f(z,c):=z*z+c;
finverseplus(z,c):=sqrt(z-c);
finverseminus(z,c):=-sqrt(z-c);
/* define z-plane ( dynamical ) */
zxMin:-2.0;
zxMax:2.0;
zyMin:-2.0;
zyMax:2.0;
/* */
PixelWidth:(zxMax-zxMin)/iXmax;
PixelHeight:(zyMax-zyMin)/iYmax;
array(Hits,iXmax,iYmax); /* 2D array of hits pixels . Hit > 0 means that point was in orbit */
/* 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])
          );
/* 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,
 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,
       /* push = add z at the end of list stack to continue iteration */
      stack:endcons(z,stack),
      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 = "miim",
 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",
 key = concat("c=",string(c),". There are ",string(NumberOfPoints)," points" ),
 label ([concat("HitLimit=",string(HitLimit)," Time=",string(t)),-1.2,-1.8]),
 xlabel     = "Z.re ",
 ylabel     = "Z.im",
 point_type    = dot,
 point_size    = 8,
 color         = black,
 points(xx,yy)
);