parabolic Julia set




Examples of parabolic case:





From period 1 to d (d-furcation) ( from parent to child component of Mandelbrot set)


How to compute c ?

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 = ( 2d - 1)

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




From period 2 to 2*d (2*d-furcation)


How to compute c ?




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)

Algorithms :

1. boundary scanning
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. IIM/J

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. DEM is inefficient here

4. Algorithm by Mark Braverman and Michael Yampolsky See book "Computability of Julia Sets " by Mark Braverman, Michael Yampolsky, Springer, 2008, page 53-60.

Main page


Autor: Adam Majewski

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

Strona utworzona przy pomocy programu: EditPlus www.editplus.com

About