File:Orbits near fixed point of fat Douady rabbit Julia set.png
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Summary
[edit]DescriptionOrbits near fixed point of fat Douady rabbit Julia set.png |
English: Orbits near fixed point of fat Douady rabbit Julia set. One can see a flower with n=3 petals and six = 2*n sepals |
Date | |
Source | own work with use of the program Mandel ver. 5.9 by Wolf Jung |
Author | Adam majewski |
Other versions | PhasePlot f(z) = e ^{1/z^2} by Elias Wegert |
Summary
[edit]This image shows discrete dynamical system :
based on complex quadratic function :[1]
where parameter c is :
It is a root point between period 1 and period 3 hyperbolic components of Mandelbrot set. It can be computed using :
- internal angle ( rotational number) = 1/3
- internal ray = 1.0
Image shows a zoom into dynamical z-plane centered at the alfa fixed point :
Colors of points :
- black = interior of Julia set
- green = exterior of Julia set
- white = forward orbit of some points of interior ( near fixed point alfa).
White cross shows fixed point alfa
One can see here that :
- some points of interior first escapes from alfa fixed point and after that fall into it
- exterior ( green points) is very thin ( width smaller then width of the pixel ) near alfa fixed point ( and its preimages)
How to do it ?
[edit]- Run program mandel by Wolf Jung [2]You are now on parameter plane ( left image) and use complex quadratic polynomial ( map) where c=0.0 ( default setting )
- Change c parameter : go to bifurcate point from period 1. Use main menu/Points/Bifurcate or key C to open input window. Enter a quotient ( = internal angle = rotational number ) = 1/3 and press enter. Now c = -0.125000000000000 +0.649519052838329 i. You can see it above parameter window. Period =10000 means here that program have not found the period because of numerical problems. Point c is a root point between period 1 and 3.
- Go to the dynamic z-plane ( right image ) : use main menu/File/To dynamics or F2 key. You are now ( yellow cross ) at the critical point : z = 0.000000000000000 +0.000000000000000 i
- Go to the alfa fixed point. Use main menu/Points/Find point or x key. Enter number 1 ( period=1 == fixed point ) and press enter. Now you are at point : z = -0.250000000000000 +0.433012701892219 i
- Zoom in using z key few times .
- increase iterations using main menu/Draw/Iterations ( max = 65 000 )
- choose few points near fixed point and draw its orbots using keys Ctrl-F ( press and do not release , because it is a slow dynamic !!! )
References
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current | 10:56, 6 July 2013 | 640 × 640 (12 KB) | Soul windsurfer (talk | contribs) | User created page with UploadWizard |
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Horizontal resolution | 37.8 dpc |
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Vertical resolution | 37.8 dpc |