This image shows discrete dynamical system :

${\displaystyle z_{n+1}=f(z_{n})}$

based on complex quadratic function :^[1]

${\displaystyle f_{c}(z)=z^{2}+c}$

where parameter c is :

${\displaystyle c=-0.125000000000000+0.649519052838329i}$

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 :

${\displaystyle z=-0.250000000000000+0.433012701892219i}$

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)

1. Run program mandel by Wolf Jung ^[2]You are now on parameter plane ( left image) and use complex quadratic polynomial ( map) ${\displaystyle f_{c}(z)=z^{2}+c}$ where c=0.0 ( default setting )
2. 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.
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
4. 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
5. Zoom in using z key few times .
6. increase iterations using main menu/Draw/Iterations ( max = 65 000 )
7. 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 !!! )

1. ↑ wikipedia : Complex_quadratic_polynomial
2. ↑ Program Mandel by Wolf Jung

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