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

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 ?

=Code=

=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= = videos=
 * Geogebra (interactive)
 * Mandelbrot Iteration Orbits by Ben Sparks
 * Mandelbrot kümesi by Hüseyin KAVAK
 * Java Script by Stefan Forcey
 * Mandelbrot Set as a Vortex Generator by Fractal Woman

=See also=
 * Spreading points on a disc

=References=