Lets see what happens with periodic z-points of complex quadratic map when c is going from c=1 toward c=-2 along horizontal axis ?
Here is 2D diagram of period 1 points ( fixed points) :
Here is 2D diagram of periodic points for periods 1-2 :
Note that :
- periodic points are superattracting when one of this points is z=0 ( then c is a center of hyperbolic component
of Mandelbrot set )
- when 2 curves intersects then c is a root point of hyperbolic component
of Mandelbrot set
- for c>-0.75 there is only 1 green line ( period 2 point). It is not an error, only 2 period 2 points have the same real part = -0.5 but different imaginary parts. See images below for more explanations
This image shoud be changed because fixed points should also show rotation of 2 parabolas)
Here one can see that both period 2 parabolas (attracting and repelling) are rotated with respect to themselvs :
Here is 2D diagram of multipliers of periocic orbits for periods 1-4
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Author: Adam Majewski