• 
  marked components

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

/*  
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)
);

1. ↑ Mandelbrot Set Chaos by JC Sprott

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