Fractals/Iterations in the complex plane/critical orbit

Definition edit

Forward orbit[1] of a critical point[2][3] is called a critical orbit.

Importance edit

Critical orbits are very important because every attracting periodic orbit[4] attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.[5][6][7]






This orbit falls into an attracting periodic cycle.

Shape edit

Relation between shape types and dynamics:

  • n-th arm spiral: attracting or repelling n-periodic orbit ( cycle)
  • closed curve: Siegel disc ( rotation)
  • n-th arm star = period n parabolic root

The shape of critical orbit can show the type of dynamics and the period

Level sets edit

Points of critical orbit ( including crirital point and fixed point = finite attractor) are on the level curves like notes on the musical staff ( dots on curves) .

How to compute attracting radius (AR) to get such effect ?

Code edit

 This software is copyright (c) Conan Dalton 2008. Permission to use it is granted under the Perl Foundations's Artistic License 2.0.
 This software includes software that is copyright (c) Paul Graham and Robert Morris, distributed under the Perl Foundations's Artistic License 2.0.
 This software uses javacc which is copyright (c) its authors
(def plot (plt c)
  (with (z 0+0i
         n 0
         repeats 0)
    (while (and (small z) (< n 10000) (< repeats 1000))
      (assign n       (+ n 1)
              z       (+ c (* z z))
              repeats (if (apply plt (complex-parts z))
                          (+ repeats 1)

images edit

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programs edit

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See also edit

References edit

  1. Wikipedia: orbit (dynamics)
  2. Wikipedia: Complex quadratic polynomial - Critical point
  3. MandelOrbits - A visual real-time trace of Mandelbrot iterations by Ivan Freyman
  4. Wikipedia: Periodic points of complex quadratic mappings
  5. M. Romera, G. Pastor, and F. Montoya: Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 10-12 (1997)
  6. Burns A M: Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104-116
  7. Khan Academy: Mandelbrot Spirals 2
  8. Complex Power Towers (Or ‘mucking around with Mathematica’) by Mike Croucher
  9. /DarkHeart by Chris King
  10. Alexandre Devert blog
  11. codeproject: Fractals-in-theory-and-practice