How to compute c ?

- choose t (proper decimal fraction) t=n/d
- w(t) = e^(i*t) ( point of unit circle)
- P(t) = w/2
- c(t) = P - P^2 ( point of boundary of main cardioid )

where t is an internal angle (in turns) of period 1 hyperbolic component of Mandelbrot set ( main cardioid)

here t = n /d

where :

d is fraction's denominator ( odd number )

n = integer from 0 to (d-1) is fraction's numerator

d is a period of child component of Mandelbrot set

angles are measured in turns modulo full turn, it means that for example 0 = 1 mod 1

denominator of external angle of rays landing on parabolic point = ( 2

From period 0 to 1

c = 1/4 ( root of period 1 component) with external rays (0/1,1/1) landing on parabolic fixed point z= beta =alfa =0.5

From period 1 to 2 (bifurcation)

parameter plane :

c = - 3/4 = - 0.75 ( root point of period 2 component) is a landing point of :

angle of internal ray = internal angle = 1/2

2 external (parameter) rays : (1/3, 2/3)

one internal ray with (internal) angle = 1/2

dynamic plane :

dynamic rays : (1/3, 2/3) land on point z = -0.5 which is period 1 parabolic cycle

Its preimages are cut points

How to compute c ?

- choose t (proper decimal fraction) t=n/d
- w(t) = e^(i*t) ( point of unit circle)
- P(t) = w/4
- c(t) = P - 1 ( point of boundary of period 2 component of Mandelbrot set )

From period 2 to 6 (trifurcation)

Internal angle of parent component of Mandelbrot set= 1/3 :

here c = exp(2*%pi*%i/3)/4 -1 = 0.21650635094611*%i-1.125

( It is a root point of period 6 component connected with period 2 component of Mandelbrot set )

Period 2 parabolic cycle :

z0:0.1703125096583*%i-1.135614939209353;

z1:0.13561493920935-0.1703125096583*%i;

with external rays : (22/63 , 25/63, 37/63 ) landing on z1

with external rays : (11/63 , 44/63, 58/63 ) landing on z0

Image of Parabolic Julia set with external rays and Maxima src code

Internal angle = 2/3 :

here c = exp(4*%pi*%i/3)/4 -1 = -0.21650635094611*%i-1.125

13/63 ,26/63 ,52/63 , 41/63, 19/63 , 38/63

C = 0.27334 + 0.00742i (Parabolic case near 20-attractor basin)

1.

ALGORITHM OF COMPUTER MODELLING OF JULIA SET IN CASE OF PARABOLIC FIXED POINT by N.B.Ampilova, E.Petrenko

Abstrect : "It is well known that a computer simulation of Julia set for a rational function with a parabolic fixed point deals with problems owing to limited precision of computations. The offered method is based on succesive iterations of small parts of Julia set. The iterations are constructed using the modified method of boundary scanning. "

2.

1. use repelling fixed point beta as a z0 of backward iteration. Note that this point is a tip point.

2. compute and draw repelling period 2 cycle. Note that these points ( and its backward iterations ) are branching points ....

3.

4.

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

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