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

is parameter c

Fc( Zcr ) = c

( 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 :

*

*

*

The fixed point is in the center of star.

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 =

* 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

c = 0.37496784+%i*0.21687214 is :

* near internal ray for angle

* 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)

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

* 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

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

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

Goes along

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

- demo 2 page 7 from program Mandel by Wolf Jung
- Fractals with Maple a set of Maple procedures used to plot a number of fractals by Filip Piękniewski
- The Julia sets trip with orbit by Evgeny Demidov
- critical orbits in Arc by Conan Dalton
- MAELSTROMS AND BUBBLE-BEINGS by LAJOS KECSKÉS
- Orbits, Mandelbrot and Julia Sets, m-furcations by Anne Burns
- M. Romera, G. Pastor, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 10-12 (1997)

Autor: Adam Majewski

adammaj1-at-o2-dot-pl

Feel free to e-mail me. (:-))

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