File:Quadratic Golden Mean Siegel Disc IIM.png
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Summary edit
DescriptionQuadratic Golden Mean Siegel Disc IIM.png |
English: Numerical aproximation of Julia set. Map is complex quadratic polynomial. Parameter c is on boundary of main cardioid of period one component of Mandelbrot set with rotational number ( internal angle ) = Golden Mean. Made with modified inverse iteration method MIIM/J with hit limit. It contains Siegel disc Polski: Numeryczna przybliżenie zbioru Julii |
Date | |
Source | Own work |
Author | Adam majewski |
Other versions |
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Licensing edit
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Summary edit
This image shows main component of Julia set and its preimages under complex quadratic polynomial[1] up to level 100008 ( new code ) or 25 ( old code).
Main component of Julia set :
- contains Siegel disc ( around fixed point )
- its boundary = critical orbit
- it is a limit of iterations for every other component of filled Julia set
computing c parameter edit
Parameter c is on boundary of period one component of Mandelbrot set ( = main cardioid ) with combinatorial rotation number ( internal angle ) = inverse Golden Mean[2] .
As a result one gets function describing relation between parameter c and internal angle :
One can compute boundary point c
of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this code by Wolf Jung[3]
phi *= (2*PI); // from turns to radians
cx = 0.5*cos(phi) - 0.25*cos(2*phi);
cy = 0.5*sin(phi) - 0.25*sin(2*phi);
Algorithm edit
- draw critical orbit = forward orbit of critical point ( "Iterates of critical point delineate a Siegel disc"[4])
- for each point of critical orbit draw all its preimages up to LevelMax if Hit<HitLimit
Critical orbit in this case is : dense [5]( correct me if I'm wrong ) in boundary of component of filled-in Julia set containing Siegel disc.[6]
Mathemathical description edit
Description [7]
Quadratic polynomial [8] whose variable is a complex number
contains invariant Siegel disc :
Boundary of Siegel disc
- contains critical point :
- is a Jordan curve
- is invariant under quadratic polynomial :
- is a closure of forward orbits of critical points
Julia is build from preimages of boundary of Siegel disc ( union of copies of B meeting only at critical point and it's preimages ):[9]
Here maximal level is not infinity but finite number :
jMax = LevelMax = 100008;
Code efficiency edit
Recursion edit
This code uses recursion[10] inside DrawPointAndItsInverseOrbit function so its runing time is (if I'm not wrong ) exponential [11] For level about 25 old code needs 24 hours and for level LevelMax new code needs 37m50.959.
Number of points edit
Level | New components | All components | New points | All points |
---|---|---|---|---|
0 | 1 | 1 | NrOfCrOrbitPoints | NrOfCrOrbitPoints |
1 | 1 | 2 | NrOfCrOrbitPoints | 2*NrOfCrOrbitPoints |
2 | 2 | 4 | ||
3 | 4 | 8 | ||
4 | 8 | 16 | ||
... | ... | ... | ... | |
100008 | ||||
j |
Components gets smaller as level increases ( but with some varioations) so nr of points to draw can be diminished.
C src code edit
Src code was formatted with Emacs
New c code : MIIM/J ( hit limit )
Postprocessing edit
Using Image Magic :
- edge detection
- resize
- convert to black and white image
- inversion
convert L100008.pgm -convolve "-1,-1,-1,-1,8,-1,-1,-1,-1" -resize 1500x1100 -threshold 5% -negate 1500e6n95.png
Compare with edit
-
Animated version
-
Average velocity - color version
-
Average velocity - gray version
-
Orbits inside Siegel Disc
-
other Siegel disc using IIM
-
other Siegel disc using DEM
References edit
- ↑ complex quadratic polynomial
- ↑ wikipedia : Golden ratio
- ↑ Mandel: software for real and complex dynamics by Wolf Jung
- ↑ Complex Dynamics by Lennart Carleson and Theodore W. Gamelin. Page 84
- ↑ Dense set in wikipedia
- ↑ Joachim Grispolakis, John C. Mayer and Lex G. Oversteegen Journal: Trans. Amer. Math. Soc. 351 (1999), 1171-1201
- ↑ A. Blokh, X. Buff, A. Cheritat, L. Oversteegen The solar Julia sets of basic quadratic Cremer polynomials, Ergodic Theory and Dynamical Systems , 30 (2010), #1, 51-65,
- ↑ wikipedia : Complex quadratic polynomial
- ↑ Building blocks for Quadratic Julia sets by : Joachim Grispolakis, John C. Mayer and Lex G. Oversteegen Journal: Trans. Amer. Math. Soc. 351 (1999), 1171-1201
- ↑ wikipedia : Recursion
- ↑ Big O notation in wikipedia
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 16:39, 17 November 2011 | 1,500 × 1,100 (27 KB) | Soul windsurfer (talk | contribs) | New version : new code and new conversion | |
08:38, 16 November 2011 | 1,500 × 1,100 (39 KB) | Soul windsurfer (talk | contribs) | 25 level - 2 days !!! ( I must improve program) | ||
13:28, 13 November 2011 | 1,500 × 1,100 (39 KB) | Soul windsurfer (talk | contribs) |
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