Fractals/Iterations in the complex plane/critical orbit

Definition

Forward orbit of a critical point is called a critical orbit.

Importance

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

$z_{0}=z_{cr}=0\,$

$z_{1}=f_{c}(z_{0})=c\,$

$z_{2}=f_{c}(z_{1})=c^{2}+c\,$

$z_{3}=f_{c}(z_{2})=(c^{2}+c)^{2}+c\,$

$...\,$

This orbit falls into an attracting periodic cycle.

Shape

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

• 
  parabolic n-th arm star
• 
  Siegel disc ( closed curve)
• 
  weakly attracting fixed point = long spiral

Level sets

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 ?

• 
  period 1 parabolic
• 
  period 1 attracting
• 
• 
  period 2 attracting

Code

"https://github.com/conanite/rainbow/blob/master/src/arc/rainbow/spiral.arc
 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)
                          0)))))

images

• images of critical orbitsat commons
• by Mike Croucher
• Chris King
• Kerry Mitchel: cardioid-boundary-orbits
• Visualizing Escape Paths in the Mandelbrot Set by Anne M. Burns
• Stefan Zenker

lists

• critical orbits
• Lori GardiThe Mandelbrot set and the fractal nature of light, the Universe, and everything by
• The Mandelbrot Set as a Quasi-Black Hole by Lori Gardi
• Mandelbrot Z Orbits by Stefan Bion
• Plot the orbits from the list by Stefan Bion
• images by Conan written in Rainbow
  • intro
  • examples
• Mandelbrot Sequences and Orbits by Stefan Forcey
• Moiré interferences in the map of orbits of the Mandelbrot Set by P. Alcover Published 2017

programs

• Geogebra (interactive)
  • Mandelbrot Iteration Orbits by Ben Sparks
  • Mandelbrot kümesi by Hüseyin KAVAK
• Java Script by Stefan Forcey

videos

• Mandelbrot Set as a Vortex Generator by Fractal Woman

See also

• Spreading points on a disc