Fractals/Iterations in the complex plane/critical orbit

Definition Forward orbitWikipedia: orbit (dynamics) of a critical pointWikipedia: Complex quadratic polynomial - Critical pointMandelOrbits - A visual real-time trace of Mandelbrot iterations by Ivan Freyman is called a critical orbit. Importance Critical orbits are very important because every attracting periodic orbitWikipedia: Periodic points of complex quadratic mappings attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.M. Romera, G. Pastor, and F. Montoya: Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 10-12 (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. 104-116 Khan Academy: Mandelbrot Spirals 2 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 "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 CroucherComplex Power Towers (Or ‘mucking around with Mathematica’) by Mike Croucher Chris King /DarkHeart by 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 Alexandre Devert blogcodeproject: Fractals-in-theory-and-practice References