File:Julia-p9.png
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Summary
[edit]DescriptionJulia-p9.png |
English: Julia set for fc(z)=z^2 + c for c= -0.031110+0.79111*i with external rays |
Source | Made with use of program mandel ver 5.5 by Wolf Jung |
Author | Adam majewski |
Long description
[edit]Parameter of quadratic map
[edit]
It is a center of period 9 hyperbolic component.
It means that critical point z=0 has period 9 under complex quadratic polynomials.
Period 9 cycle
[edit]Here is superattractive orbit of critical point :
z0 = 0.000000000000000 +0.000000000000000 i
z1 = -0.031467614571113 +0.790703205755478 i
z2 = -0.655688963396307 +0.740940118317765 i
z3 = -0.150531856784133 -0.180949310481546 i
z4 = -0.041550427627981 +0.845180477136670 i
z5 = -0.744071215468013 +0.720467985259779 i
z6 = 0.003100241332648 -0.281455773240590 i
z7 = -0.110675355365251 +0.788958044112452 i
z8 = -0.641673375655634 +0.616066781954641 i
z9 = 0.000738826626237 +0.000075902543203 i
abs(z0-z9) = 7.427152749886263*10^-4
Points are marked on image with white points
Period 3 cycle
[edit]In the Julia set there is a 3-period repelling orbit :
Here absolute value of multiplier is > 1 ( numerical computations give 1.11).
This points are the landing points for rays listed below.
Parabolic orbit portrait
[edit]Orbit portrait consist of 3 list, each has 3 angles :
These are angles of external rays landing of 3 points of parabolic cycle
Note that denominator of angles is :
Period
[edit]Here are 5 values :
- orbit period = period of orbit from orbit portrait under complex quadratic map = p = 3
- period of external angle under doubling map = r = 3
- ray period = period of point z on ray under complex quadratic map = r*p = 9
- period of landing point under complex quadratic map = 3
- period of orbit of critical point = 9
How to use Mandel
[edit]Keys :
- e for externalray
- f for forward orbit
- s for description of angle and move to landing point
To check period of ray :
- go to the point on ray ( exterior of filled Julia set) near Julia set
- apply forward iteration using f key
To check period of landing point of ray :
- draw ray ( press e key and enter angle )
- press s key, enter the same angle and click OK
- * apply forward iteration using f key
Licensing
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
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