Critical point of complex quadratic mappings





Critical point of Mandelbrot function = Zcr
is a point of dynamical plane where the derivative is equal to zero :
F'c(Zcr)=0
2 * Zcr = 0
Zcr = 0
So 0 is the only critical point of Mandelbrot function

Critical value of Fc(z) = value of Fc(z) at Critical point
is parameter c
Fc( Zcr ) = c

Critical orbit is a forward orbit of critical point z = 0 is
( 0, C, CC+C, (CC+C)(CC+C)+C, ...)

Note thet critical orbit is a set of points made by iteration of one ( critical) point. Even if it can be devided into subsets ( like spirals) it is still one set ( path).




If internal angle of c is :
* greater then angle of closest rational internal ray then forward orbit of critical point is turning around fixed point in the counterclockwise direction instead of going straight to it.
* smaler then angle of closest rational internal ray then orbit will be turning around fixed point in the clockwise direction
* equal to an angle of closest internal ray with rational angle then orbit looks like n-arms star, where n is internal angle denominator.

The fixed point is in the center of star.



Intresting c values :

Inside main cardioid



i/3


c = -0.14222874119878 -0.64732701703906*i
It's multiplier of fixed point z = alfa has :
* absolute value = 0.99777759940413. It means it is inside main cardioid ( abs value < 1).
* angular part = 0.66349828489868
* points = 500
* z plane : zx =(-0.55,-0,05) ; zy = (-0.65,-0.15)

So it is :
* near internal ray for angle = 2/3 = 0.666666...
* turning around fixed point in the clockwise direction ( angular part < 2/3 )
* near root point of period 3 hyperbolic component with internal adress : 1 -> 3

One can see that it forms 3 spirals tending to weakly atracting fixed point z= -0.25799546044574 -0.42699927032439*%i

Similar image will give c point near root point for internal angle 1/3


i/6




c = 0.37496784+%i*0.21687214 is :
* near internal ray for angle 1/6
* forward orbit of critical point is turning around fixed point in the counterclockwise direction instead of going straight to it (angular part > 1/6 )
* points = 100 000
* z plane : zx =(-0.1 , 0.5) ; zy = (0.0, 0.7)




i/7



c = 0.1144+0.5956i
It's multiplier of fixed point z = alfa has :
* absolute value = 0.99984647669665. It means it is inside main cardioid.
* angular part = 0.28560795866178

So c on paramete plane is :
* near internal ray for angle 2/7 = 0.28571428571429
* near root point of period 7 hyperbolic component with internal adress : 1 -> 7

One can see that it forms 7 spirals tending to weakly atracting fixed point z = -0.11091774979951 + 0.48746332889777*%i

Similar image will give c near root point for internal angle 1/7, 3/7, 4/7, 5/7 and 6/7

c:0.6*%i+0.1
It's multiplier of fixed point z = alfa has :
* absolute value = 0.99719727009848. It means it is inside main cardioid ( its value < 1).
* angular part = 0.28864000669183

So it is :
* near internal ray for angle 2/7 = 0.28571428571429
* near root point of period 7 hyperbolic component with internal adress : 1 -> 7

One can see that it forms 7 spirals tending to weakly atracting fixed point z=-0.11986524496968 + 0.48397615842242*%i


mandelbrot-spiral-core-7 by conanite at flicker
mandelbrot-spiral-on-the-cusp-of-7 by conanite at flicker
22-explosion by conanite at flicker
22-spirals by conanite at flicker


Outside main cardioid


c = -0.50977517291904 -0.60039090737700*i
It's multiplier of fixed point z = alfa has :
* absolute value =1.074821068645567. It means it is outside main cardioid ( its value > 1).
* angular part = 0.6026014183285282
* points = 5000
* z plane : zx =(-0.90 , 0.15) ; zy = (-0.62, 0.8), dx=1.05, dy = 1.42
* z = alpha = -0.429553255941040 -0.322945943946584*%i is near center of image

So it is outside main cardioid but inside period 5 hyperbolic component with : * internal adress : 1 -> 5
* angled internal adress : 1-(3/5)-> 5
* with root point on internal ray of main cardioid for angle 3/5

Periodic points z, stability index =abs(m(z) and intenal angle = arg(m(z) :
z=.09416549555568843*%i+.05706137343847581; abs(multiplier(z))=.9737268947882353; arg(m(z)) = 0.576738755466852
z=.007398484272453143*%i-0.591832736734896; abs(multiplier(z))=.9737268947883195; arg(m(z)) = .5767387554667789
z=-.6091482377662949*%i-.1595639222174411; abs(multiplier(z))=0.973726894788254; arg(m(z)) = .5767387554668595
z=-.5896444823631454*%i-.5153863131336123; abs(multiplier(z))=.9737268947877692; arg(m(z)) = .5767387554667908
z=-.4059947433168776*%i-.8553761032192347; abs(multiplier(z))=.9737268947813147; arg(m(z)) = .5767387554661696




/* Maxima CAS program */
/* argument of complex number in turns */
carg_t(z):=
 block(
 [t],
 t:carg(z)/(2*%pi),  /* now in turns */
 if t<0 then t:t+1, /* map from (-1/2,1/2] to [0, 1) */
 return(t)
)$


/* square root of complex number */
csqrt(z):=
 block(
  [t,w,re,im],
  t:abs(z)+realpart(z),
  if t>0 
   then (re:sqrt(t/2), im:imagpart(z)/sqrt(2*t))
   else  (im:abs(z), re:0),
  w:re+im*%i,
  return(w)
)$



c: -0.14222874119878-0.64732701703906*%i;
l:1-csqrt(1-4*c); /* multiplier of fixed point */
r:cabs(l); /* radius */
a:carg_t(l); /* angle */
fixed:l/2; /* alfa */



This animated image shows how critical orbit changes.
Trip starts from c=0. It is a center of period one component ( = main cardioid).
Goes along internal ray of angle =1/6 of main cardioid.



Image and src code
# angle = 0.16666667 = 1/6
# Cx         Cy      		radius inside main cardioid
 0.00000000  0.00000000  # 0.00000000 = center of main cardioid
 0.02625000  0.04113621  # 0.10000000 
 0.05500000  0.07794229  # 0.20000000 
 0.08625000  0.11041824  # 0.30000000 
 0.12000000  0.13856406  # 0.40000000 
 0.15625000  0.16237976  # 0.50000000 
 0.19500000  0.18186533  # 0.60000000 
 0.23625000  0.19702078  # 0.70000000 
 0.28000000  0.20784610  # 0.80000000 
 0.32625000  0.21434129  # 0.90000000  ? 0.37496784+0.21687214*i ? 
 0.37500000  0.21650635  # 1.00000000 = root point of period 6 hyperbolic component
# Cx         Cy      		t ( radius inside period 6 hyperbolic component 
 0.37359932  0.21657192  # 0.10000000 
 0.37219863  0.21663749  # 0.20000000 
 0.37079795  0.21670306  # 0.30000000 
 0.36939726  0.21676863  # 0.40000000 
 0.36799658  0.21683420  # 0.50000000 
 0.36659590  0.21689977  # 0.60000000 
 0.36519521  0.21696534  # 0.70000000 
 0.36379453  0.21703091  # 0.80000000 
 0.36239384  0.21709648  # 0.90000000 
 0.36099316  0.21716205  # 1.00000000 
 0.35959248  0.21722762  # 1.10000000 





Main page


Autor: Adam Majewski
adammaj1-at-o2-dot-pl
Feel free to e-mail me. (:-))

About
fraktal.republika.pl