# Fractals/Iterations in the complex plane/Mandelbrot set/centers
## name ( synonims )

* center of hyperbolic component of Mandelbrot set
* Nucleus of a Mu-Atom

## definition

A center of a hyperbolic component H is a parameter $c_{0}\in H\,$ ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0."

## Initial domain

* "All centers of M of any period are contained in the interior of M, so that the circle C = {z ∈ C: |z − 0.75| = 2} surrounds all centers. "
* "one can exploit the real symmetry of M and only use starting points with non-negative imaginary parts; "

## Period

Period of center = period of critical orbit = period of hyperbolic component

## Julia set

It is the easiest case for drawing Julia sets.
Dynamical plane consist of 2 superattracting basins
* exterior ( ininity is a superattracting fixed point)
* interior ( z=0 is one of superattracting periodic points)
* period 33 center
See commons categories:
* commons Category:Circle Julia sets = period 1 center Julia set
* commons Category:Basilica Julia sets = period 2 Julia set
* period 3 centers
** commons Category:Douady rabbit Julia sets
** commons Category:Airplane Julia sets

## Number of Mandelbrot set components

Equation :
$N(p)=2^{p-1}-s$
where :
* N(p) is a number of all components of Mandelbrot set for given period p.
* p is a period
* s is a sum of numbers of its components for each divisor of period smaller than period p
All values are positive integers. It is sequence A000740
Program in Maxima CAS :
One can run it :
or :
result :
Also c program:
See also
* Counting Hyperbolic Components in the Main Molecule

## How to compute centers ?

Methods :
* all centers for given period $p\,$
** solving $F_{p}(0,c)=0\,$ or $G_{p}(0,c)=0\,$
*** showing results in graphical form
*** using numerical methods
**** "quick and dirty" algorithm : check if $abs(z_{n})<eps$ then colour c-point with colour n. Here n is a period of attracting orbit and eps is a radius of circle around attracting point = precision of numerical comutations
**** finding all roots of polynomial
***** The iterated refinement Newton method
***** the Ehrlich-Aberth iteration ( Mpsolve )
**** atom domains
**** Myrberg's method
*** interval arithmetics and a divide and conquer strategy
** "Searching is performed using a quadtree subdivision of the grid which increases search resolution in areas of higher root density. Within a search area first a map of the function magnitude is made at moderate resolution, then points of local minimum are refined using 640 bit fixed point math. Root searching requires twice the precision of the desired result due to multiplying very small numbers. As roots are added the quadtree is subdivided further and the search continues. Only the upper half plane is searched due to symmetry.
Quadtree depth was purposely limited on higher periods to reduce run time since the number of roots becomes unmanageable." James Artherton
* one center $c_{p}\,$ for given period $p\,$ near given point $c\,$
** some steps of Newton method for solving $F_{p}(0,c_{p})=0\,$

#### System of equations defining centers

Center of hyperbolic component is a point $c\,$ of parameter plain, for which periodic z-cycle is superattracting. It gives a system of 2 equations defining centers of period $p\,$ hyperbolic components :
* first defines periodic point $z_{0}\,$,
* second makes $z_{0}\,$ point superattracting.
${\begin{cases}F_{p}(z_{0},c)=z_{0}\\{\frac {df_{c}^{(p)}(z_{0})}{dz}}=0\end{cases}}$
see definitions for details.
"These polynomials have integer coefficients. They can be obtained by a recurrent relation on the degree. Let Pd the polynomial of degree d. We have"

#### Solving system of equations

Second equation can be solved when critical point $z_{cr}=0\,$ is in this cycle :
$z_{0}=z_{cr}=0\,$
To solve system put $z_{0}\,$ into first equation.

##### Equation defining centers

One gets equation :
$F_{p}(0,c)=0\,$
Centers of components are roots of above equation.
Because $z=0\,$ one can remove z from these equations :
* for period 1 : z^2+c=z and z=0 so $c=0$
* for period 2 : (z^2+c)^2 +c =z and z=0 so $c^{2}+c=0$
* for period 3 : ((z^2+c)^2 +c)^2 +c = z and z=0 so $(c^{2}+c)^{2}+c=0$
Here is Maxima functions for computing above functions :

##### Reduced equation defining centers

Set of roots of above equation contains centers for period p and its divisors. It is because :
$F_{p}(0,c)=\prod _{m|p}G_{m}(0,c)\,$
where :
* $G\,$ = Gleason's period-m polynomial= irreducible divisors of $F_{p}\,$
* capital Pi notation of iterated multiplication is used
* $m|p\,$ means here : for all divisors of $p\,$ ( from 1 to p ). See table of divisors.
For example, :
$F_{1}=c=G_{1}\,$
$F_{2}=c*(c+1)=G_{1}*G_{2}\,$
$F_{3}=c*(c^{3}+2*c^{2}+c+1)=G_{1}*G_{3}\,$
$F_{4}=c*(c+1)*(c^{6}+3*c^{5}+3*c^{4}+3*c^{3}+2*c^{2}+1)=G_{1}*G_{2}*G_{4}\,$
So one can find irreducible polynomials using :
$G_{p}(0,c)={\frac {F_{p}(0,c)}{\prod _{m|p,m<p}G_{m}(0,c)}}\,$
Here is Maxima function for computing $G\,$ :
Here is a table with degree of $F\,$ and $G\,$ for periods 1-10 and precision needed to compute coefficients of these functions (the roots can be calculated with lower precision, provided you work with the unexpanded form $(\ldots (c^{2}+c)^{2}+c\ldots )^{2}+c$).
| period $p\,$ | $\deg(F_{p}(0,c))\,$ | $\deg(G_{p}(0,c))\,$ | $fpprec\,$ |
| 1 | 1 | 1 | 16 |
| 2 | 2 | 1 | 16 |
| 3 | 4 | 3 | 16 |
| 4 | 8 | 6 | 16 |
| 5 | 16 | 15 | 16 |
| 6 | 32 | 27 | 16 |
| 7 | 64 | 63 | 32 |
| 8 | 128 | 120 | 64 |
| 9 | 256 | 252 | 128 |
| 10 | 512 | 495 | 300 |
| 11 | 1024 | 1023 | 600 |
Where:
* fpprec is the number of significant decimal digits for arithmetic on bigfloat numbers
Here is a table of greatest coefficients.
| period $p\,$ | $log10(gc_{p})\,$ | $gc_{p}\,$ |
| 1 | 0 | 1 |
| 2 | 0 | 1 |
| 3 | 1 | 2 |
| 4 | 1 | 3 |
| 5 | 3 | 116 |
| 6 | 4 | 5892 |
| 7 | 11 | 17999433372 |
| 8 | 21 | 106250977507865123996 |
| 9 | 44 | 16793767022807396063419059642469257036652437 |
| 10 | 86 | 86283684091087378792197424215901018542592662739248420412325877158692469321558575676264 |
| 11 | 179 | 307954157519416310480198744646044941023074672212201592336666825190665221680585174032224052483643672228833833882969097257874618885560816937172481027939807172551469349507844122611544 |
Precision can be estimated as bigger than size of binary form of greatest coefficient $log_{2}(gc)\,$ :
| period $p\,$ | $\deg(G_{p}(0,c))\,$ | $\log _{2}(gc_{p})\,$ | $fpprec\,$ |
| 1 | 1 | 1 | 16 |
| 2 | 1 | 1 | 16 |
| 3 | 3 | 2 | 16 |
| 4 | 6 | 2 | 16 |
| 5 | 15 | 7 | 16 |
| 6 | 27 | 13 | 16 |
| 7 | 63 | 34 | 32 |
| 8 | 120 | 67 | 64 |
| 9 | 252 | 144 | 128 |
| 10 | 495 | 286 | 300 |
| 11 | 1023 | 596 | 600 |
Here is Maxima code for $gc\,$
See also:
* math.stackexchange question: irreducible-factors-of-mandelbrot-polynomials

#### Graphical methods for finding centers

All these methods shows centers for period n and its divisors.

##### color is proportional to magnitude of zn

* color is proportional to magnitude of zn
* Parameter plane is scanned for points c for which orbit of critical point vanishes
* YouTube video : Mandelbrot Oscillations

##### color shows in which quadrant zn lands

This is radial nth-decomposition of exterior of Mandelbrot set ( compare it with n-th decomposition of LSM/M )
4 colors are used because there are 4 quadrants :
* re(z_n) > 0 and im(z_n) > 0 ( 1-st quadrant )
* re(z_n) < 0 and im(z_n) > 0 ( 2-nd quadrant )
* re(z_n) < 0 and im(z_n) < 0 ( 3-rd quadrant )
* re(z_n) > 0 and im(z_n) < 0 (4-th quadrant ).
"... when the four colors are meeting somewhere, you have a zero of q_n(c), i.e., a center of period dividing n. In addition, the light or dark color shows if c is in M." ( Wolf Jung )
Here is fragment of c code for computing 8-bit color for point c = cx + cy * i :
One can also find cpp code in function quadrantqn from class mndlbrot in mndlbrot.cpp file from source code of program Mandel by Wolf Jung
The differences from standard loop are :
* there is no bailout test (for example for max value of abs(zn) ). It means that every point has the same number of iterations. For large n overflow is possible. Also one can't use test Iteration==IterationMax
* Iteration limit is relatively small ( start for example IterationMax = 8 )
See also code in Sage

##### using Gaussian wave function method
### Newton method

Newton method for finding one root

## Lists of centers

* centers for periods 1-20 ( sum = 1 045 239 ) by James Artherton. Made with custom software using 320 bit fixed point math (approx. 92 decimal digits) and a quadtree subdivision of the grid to increase the search resolution in areas with higher root density. This adapts to very fine resolution in the tips of the antennae.
* largest islands by Robert P Munafo
* a database of all islands up to period 16, found by tracing external rays (period, islandhood, angled internal address, lower external angle numerator, denominator, upper numerator, denominator, orientation, size, centre realpart, imagpart)
* real centers of components computed by Jay R. Hill
* collection of centers lists
* islands
* madore.org/~david math/mandpoints.dat
List:
* c = 0.339410819995598 -0.050668285162643 i period = 11, distorted minibrot
* c = 0.325589509550660 -0.038047880934756 i period = 12, distorted minibrot
* c = 0.314559489984000 -0.029273969079000 i period = 13, distorted minibrot
* c = 0.272149607027528 +0.005401159465460 i period = 22, distorted minibrot

## programs
### MPSolve
#### mandelbrot-solver

It is a package for Ubuntu A solver for Mandelbrot polynomials based on MPSolve.
A polynomial rootfinder that can determine arbitrary precision approximations with guaranteed inclusion radii. It supports exploiting of polynomial structures such as sparsisty and allows for polynomial implicitly defined or in some non standard basis. This binary, provided as an example of custom polynomial implementation in the MPSolve package, uses the particular structure of this family of polynomials to develop an efficient solver that exhibit experimental O(n^2) complexity.
This package contains the specialization of mpsolve to Mandelbrot polynomials.
example :
It seems that:
and then results are centers.

## Notes

* ^ It is conjectured that the polynomials (for the centers) are irreducible, but this has not been proved yet - Wolf Jung

## References

* ↑ Nucleus - From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.
* ↑ Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
* ↑ NEWTON’S METHOD IN PRACTICE: FINDING ALL ROOTS OF POLYNOMIALS OF DEGREE ONE MILLION EFFICIENTLY by DIERK SCHLEICHER AND ROBIN STOLL
* ↑ divisor in wikipedia
* ↑ FF Topic: The Mandelbrot Polynomial Roots Challenge (Read 5233 times)
* ↑ Piers Lawrence and Rob Corless : Mandelbrot Polynomials and Matrices. 1 ; 048 ; 575 roots of period 21
* ↑ IAP Poster 2013
* ↑ Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees by Robin Stoll, Dierk Schleicher
* ↑ wikipedia : MPSolve
* ↑ Determination of Mandelbrot Set's Hyperbolic Component Centres by G Alvarez, M Romera, G Pastor and F Montoya. Chaos, Solitons & Fractals 1998, Vol 9 No 12 pp 1997-2005
* ↑ Myrberg, P J, Iteration der reelen polynome zweiten GradesIII, Ann. Acad. Sci. Fennicae, A. I no. 348, 1964.
* ↑ evaldraw script that can compute 1 million root by knighty
* ↑ A ROOTFINDING ALGORITHM FOR POLYNOMIALS AND SECULAR EQUATIONS by LEONARDO ROBOL
* ↑ mathoverflow question: discriminants-of-gleasons-period-n-polynomials-for-the-mandelbrot-set
* ↑ irreducible divisors at wikipedia
* ↑ V Dolotin , A Morozow : On the shapes of elementary domains or why Mandelbrot set is made from almost ideal circles ?
* ↑ maxima manual : fpprec
* ↑ Mandelbrot Set in R by J.D. Long
* ↑ Fractal Stream help file
* ↑ youtube video : Mandelbrot Oscillations
* ↑ Multiplatform cpp program Mandel by Wolf Jung
* ↑ Formation of Escape-Time Fractals By Christopher Olah
* ↑ _3__The_Dark Exploding the Dark Heart of Chaos by Chris King March-Dec 2009
* ↑ Fractal Dragons by James Artherton
* ↑ Largest Islands by Robert P. Munafo, 2010 Oct 21
* ↑ mrob : largest-islands.txt
* ↑ a database of all islands up to period 16, found by tracing external rays by Claude
* ↑ real centers of components computed by Jay R. Hill
* ↑ mset centers
* ↑ math.stackexchange question: mini-mandelbrots-are-they-exact-copies?
* A comparison of solution methods for Mandelbrot-like polynomials by Eunice Y. S. Chan. 2016