Fractals/Iterations in the complex plane/critical orbit
< Fractals
Definition
editForward orbit^{[1]} of a critical point^{[2]}^{[3]} is called a critical orbit.
Importance
editCritical 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
editRelation between shape types and dynamics:
 nth arm spiral: attracting or repelling nperiodic orbit ( cycle)
 closed curve: Siegel disc ( rotation)
 nth arm star = period n parabolic root
The shape of critical orbit can show the type of dynamics and the period

parabolic nth arm star

Siegel disc ( closed curve)

weakly attracting fixed point = long spiral
Level sets
editPoints 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
edit"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 (complexparts z))
(+ repeats 1)
0)))))
images
edit images of critical orbitsat commons
 by Mike Croucher^{[8]}
 Chris King ^{[9]}
 Kerry Mitchel: cardioidboundaryorbits
 Visualizing Escape Paths in the Mandelbrot Set by Anne M. Burns
 Stefan Zenker
lists
edit critical orbits
 Lori GardiThe Mandelbrot set and the fractal nature of light, the Universe, and everything by
 The Mandelbrot Set as a QuasiBlack 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
 Mandelbrot Sequences and Orbits by Stefan Forcey
 Moiré interferences in the map of orbits of the Mandelbrot Set by P. Alcover Published 2017
programs
editvideos
editSee also
edit Spreading points on a disc ^{[10]}^{[11]}
References
edit ↑ Wikipedia: orbit (dynamics)
 ↑ Wikipedia: Complex quadratic polynomial  Critical point
 ↑ MandelOrbits  A visual realtime trace of Mandelbrot iterations by Ivan Freyman
 ↑ Wikipedia: Periodic points of complex quadratic mappings
 ↑ M. Romera, G. Pastor, and F. Montoya: Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 1012 (1997)
 ↑ Burns A M: Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104116
 ↑ Khan Academy: Mandelbrot Spirals 2
 ↑ Complex Power Towers (Or ‘mucking around with Mathematica’) by Mike Croucher
 ↑ /DarkHeart by Chris King
 ↑ Alexandre Devert blog
 ↑ codeproject: Fractalsintheoryandpractice