# Fractals/Iterations in the complex plane/parabolic

"Most programs for computing Julia sets work well when the underlying dynamics is hyperbolic but experience an exponential slowdown in the parabolic case." (Mark Braverman)[1]

In other words it means that one can need days for making a good picture of parabolic Julia set with standard / naive algorithms.

There are 2 problems here:

• slow (= lazy) local dynamics (in the neighbourhood of a parabolic fixed point)
• some parts are very thin (hard to find using standard plane scanning)

# Planes

## Dynamic plane

Discrete dynamic in case of complex quadratic polynomial. exterior = white, interior = gray, unknown=red;

How image is changing with various Iteration Max

Dynamic plane = complex z-plane ${\displaystyle \mathbb {C} =J_{f}\cup F_{f}}$ :

• Julia set ${\displaystyle J_{f}\subset \mathbb {C} }$  is a common boundary: ${\displaystyle J_{f}\,=\partial A_{f}(p)=\partial A_{f}(\infty )}$
• Fatou set ${\displaystyle F_{f}\subset \mathbb {C} }$
• exterior of Julia set = basin of attraction to infinity: ${\displaystyle A_{f}(\infty )\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to \infty \ as\ k\to \infty \}.}$
• interior of Julia set = basin of attraction of finite, parabolic fixed point p: ${\displaystyle A_{f}(p)\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to p\ as\ k\to \infty \}.}$
• immediate basin = sum of components which have parabolic fixed point p on it's boundary  ; the immediate parabolic basin of p is the union of periodic connected components of the parabolic basin.
• attracting Lea-Fatou flower = sum of n attracting petals = sum of 2*n attracting sepals
• petal = part of the flower. Each petal contains part of 2 sepals,
• sepals (Let 1 be an invariant curve in the first quadrant and L 1 the region enclosed by 1 ∪ {0}, called a sepal.)[2]

${\displaystyle \mathbb {C} \supset F(f)\supset A_{f}(p)}$

See also:

• Filled Julia set[3] ${\displaystyle K_{f}}$

# Key words

• parabolic chessboard or checkerboard
• parabolic implosion
• Fatou coordinate
• Hawaiian earring
• in wikipedia = "The plane figure formed by a sequence of circles that are all tangent to each other at the same point and such that the sequence of radii converges to zero." (Barile, Margherita in MathWorld)[4]
• commons:Category:Hawaiian earrings
• Gevrey symptotic expansions
• the Écalle-Voronin invariants of (7.1) at the origin which have Gevrey- 1/2 asymptotic expansions[5]
• germ[6][7][8]
• germ of the function: Taylor expansion of the function
• multiplicity[9]
• Julia-Lavaurs sets
• The Leau-Fatou flower theorem:[10] repelling or attracting flower. Flower consist of petals
• Leau-Fatou flower
• Parabolic Linearization theorem
• The horn map
• Blaschke product
• Inou and Shishikura's near parabolic renormalization
• complex polynomial vector field[11]
• numbers
• "a positive integer ν, the parabolic degeneracy with the following property: there are νq attracting petals and νq repelling petals, which alternate cyclically around the fixed point."[12]
• combinatorial rotation number
• a Poincaré linearizer of function f at parabolic fixed point[13]
• "the parabolic pencil. This is the family of circles which all have one common point, and thus are all tangent to each other, either internally or externally."[14]

## Leau-Fatou flower theorem

The Leau-Fatou flower theorem states that, if function ${\displaystyle f^{n}}$  has the Taylor expansion[15]

${\displaystyle f^{n}(z)=z+az^{n+1}+O(z^{n+2})}$

then the complex number ${\displaystyle v}$  on the unit cirle ${\displaystyle v\in S^{1}}$  describes unit vector ( direction):

• an repulsion vector (from origin to v) for f at the origin if ${\displaystyle nav^{n}=1}$
• an attraction vector ( from v to origin ) if ${\displaystyle nav^{n}=-1}$

There are n equally spaced attracting directions, separated by n equally spaced repelling directions. The integer n+1 is called the multiplicity of fixed point

### Examples

Function expansion: z+z^5

/* Maxima CAS */
display2d:false;
taylor(z+z^5, z,0,5);

z + z^5


So :

• a = 1
• n = 4

Compute attracting directions:

/* Maxima CAS */
solve (4*v^4 = 1, v);

[v = %i/sqrt(2),v = -1/sqrt(2),v = -%i/sqrt(2),v = 1/sqrt(2)]


Function m*z+z^2

(%i7) taylor(m*z+z^2, z,0,5);
(%o7) m*z+z^2



So :

• a = 1
• n = 1
(%i9) solve (v = 1, v);
(%o9) [v = 1]


## Ecalle cylinder

Ecalle cylinders[16] or Ecalle-Voronin cylinders (by Jean Ecalle[17][18])[19]

"... the quotient of a petal P under the equivalence relation identifying z and f (z) if both z and f (z) belong to P. This quotient manifold is called the Ecalle cilinder, and it is conformally isomorphic to the infinite cylinder C/Z"[20]

## eggbeater dynamics

Physical model: the behaviour of cake when one uses eggbeater.

The mathematical model: a 2D vector field with 2 centers (second-order degenerate points)[21][22]

The field is spinning about the centers, but does not appear to be diverging.

Maybe better description of parabolic dynamics will be Hawaiian earrings

## parabolic germ

Germ:[23][24][25]

• z+z^2
• z+z^3
• z+z^{k+1}
• z+a_{k+1}z^{k+1}
• z+a_{k+1}z^{k+1}
• "a germ ${\displaystyle f(x)=a_{1}x+a_{2}x^{2}+\ldots }$  with ${\displaystyle \vert a_{1}\vert \neq 0,1}$  is holomorphically conjugated to its linear part ${\displaystyle g(x)=a_{1}z}$ " (Sylvain Bonnot)[26]

germ of vector field

## The horn map

"the horn map h = Φ ◦ Ψ, where Φ is a shorthand for Φattr and Ψ for Ψrep (extended Fatou coordinate and parameterizations)."[27]

## Petal

repelling petals around fixed point and its preimages

"The petals ... are interesting not only because they give a rather good intuitive idea of the dynamics that arise near a parabolic point, but also because that the dynamic of f0 on a petal can be linearized, i.e., it is conjugated to the shift map T of C defined by T(w) := w + 1." (Laurea Triennale[28])

There is no unified definition of petals.

Petal of a flower can be:

• attracting / repelling
• small/alfa/big/ (small attracting petals do not ovelap with repelling petals, but big do)

Each petal is invariant under f^period. In other words it is mapped to itself by f^period.

Attracting petal P is a:

• Each petal is invariant under ${\displaystyle f^{n}}$ . In other words it is mapped to itself by ${\displaystyle f^{n}}$ : ${\displaystyle f^{n}({\overline {P}})\subset P\cup \left\{p\right\}}$
• domain (topological disc) containing:
• parabolic periodic point p in its boundary ${\displaystyle {\overline {P}}\ni \left\{p\right\}}$  (precisely in its root, which is a common points of all attracting and repelling petals = center of the Lea-Fatou flower)
• critical point or it's n=period images on the other side (only small ??)
• trap which captures any orbit tending to parabolic point[29]
• set contained inside component of filled-in Julia set. The attracting petals of parabolic fixed point are contained in it's basin of attraction
• " ... is maximal with respect to this property. This preferred petal P max always has one or more critical points on its boundary."[30]
• "an attracting petal is the interior of an analytic curve which lies entirely in the Fatou set, has the right tangency property at the fixed point, and is mapped into its interior by the correct power of the map" Scott Sutherland
• "... an attracting petal is a set of points in a sufficient small disk around the periodic point whose forward orbits always remain in the disk under powers of return map." (W P Thurston: On the geometry and dynamics of Iterated rational maps)

Petals ${\displaystyle P_{j}}$  are symmetric with respect to the d-1 directions:

${\displaystyle arg(z)={\frac {2\Pi l}{d-1}}}$

where:

• d is (to do)
• l is from 0 to d-2

Petals can have different size.

If ${\displaystyle \lambda ^{n}=1}$  then Julia set should approach parabolic periodic point in n directions, between n petals.[31]

"Using the language of holomorphic dynamics, people would say that you are studying the dynamics of a polynomial near the parabolic fixed point ${\displaystyle z=0}$ . By a simple linear change of variables, the study of any such parabolic fixed point can be reduced to the study of ${\displaystyle z\mapsto z+z^{2}+O(z^{3})}$ . Then you can apply another change ${\displaystyle w=-{\frac {1}{z}}}$ . Thus you are reduced to the study of ${\displaystyle f(w)=w+1+O(1/w)}$ . If the real part $Re(w)$ is large enough you will obtain ${\displaystyle f^{n}(w)=w+n+O(\log n)}$ . This will give you what you want (when going back to the z-variable).

The domain ${\displaystyle Re(w)>R}$  (for large ${\displaystyle R}$ ) looks like some kind of cardioid (in your particular case) when you visualize it in the z-variable (it's poetically called an attracting petal)." Sylvain Bonnot[32]

### Constructing the petal

#### 0/1 or 1/1

First attracting petal is a circle with:

• center on the x axis
• radius = (cabs(zf) + cabs(z))/2

Other (bigger) attracting petals are preimages of first petal which include first petal.

where:

• z is the last point of critical orbit.
• zf is a fixed point.

#### 1/2

circular attracting petal:

• center is between fixed point and critical point.
• radius smaller than half of the distance between fixed and critical point.
// choose such value that level sets cross at z=0
// choose radius such a
double GivePetalRadius(complex double c, complex double fixed, int n){
complex double z = 0.0; // critical point
int k;
// best for n>1
int kMax = (n*ChildPeriod)  - 1; // ????

for(k=0;  k<kMax-1; ++k)
z = z*z + c; // forward iteration

return  cabs(z-fixed)/2.0;

}


#### 1/3

Method by Scott Sutherland:[33]

• choose the connected component containing the critical point
• find an analytic curve which lies entirely in the Fatou set, has the right tangency property at the fixed point, and is mapped into its interior by the correct power of the map
• the other two are just the image of this curve under the first and second iterates.

### Number of petals

In parabolic point child period coincides with parent period

For quadratic polynomials:

Multiplicity = ParentPeriod + ChildPeriod

NumberOfPetals = multiplicity - ParentPeriod

It is because in parabolic case fixed point coincidence with periodic cycle. Length of cycle (child period) is equal to number of petals.

f(z) number of petals explanation
${\displaystyle z^{d}+z}$  d-1 for ${\displaystyle z^{d}+z=z}$  point z=0 has multiplicity d
${\displaystyle z^{d}-z}$  d+2 (?)for ${\displaystyle f^{2}(z)}$  a root z=0 has multiplicity d+3

For f(z)= -z+z^(p+1) parabolic flower has:

• 2p petals for p odd
• p petals for p even[34]

... (to do)

## Sepal

Sepals and petals

Parabolic sepals for internal angle 1 over 1

Definitions:

## Flower

Sum of all petals creates a flower[36] with center at parabolic periodic point.[37]

## Cauliflower

Cauliflower or broccoli:[38]

• empty (its interior is empty) for c outside Mandelbrot set. Julia set is a totally disconnected.
• filled cauliflower for c=1/4 on boundary of the Mandelbrot set. Julia set is a Jordan curve (quasi circle).

Please note that:

• size of image differs because of different z-planes.
• different algorithms are used so colours are hard to compare.

### Bifurcation of the Cauliflower

How Julia set changes along real axis (going from c=0 thru c=1/4 and further):

Perturbation of a function ${\displaystyle f(z)}$  by complex ${\displaystyle \epsilon }$ :

${\displaystyle g(z)=f(z)+\epsilon }$

When one add epsilon > 0 (move along real axis toward + infinity) there is a bifurcation of parabolic fixed point:

• attracting fixed point (epsilon<0)
• one parabolic fixed point (epsilon = 0)
• one parabolic fixed point splits up into two conjugate repelling fixed points (epsilon > 0)

"If we slightly perturb with epsilon<0 then the parabolic fixed point splits up into two real fixed points on the real axis (one attracting, one repelling)."

See:

• demo 2 page 9 in program Mandel by Wolf Jung

## parabolic implosion

Video on YouTube[39]

## Vector field

• 2D vector field and its

### singularity

singularity types:

• center type: "In this case, one can find a neighborhood of the singular point where all integral curves are closed, inside one another, and contain the singular point in their interior"[40]
• non-center type: neighborhood of singularity is made of several curvilinear sectors:[41]

"A curvilinear sector is defined as the region bounded by a circle C with arbitrary small radius and two streamlines S and S! both converging towards singularity. One then considers the streamlines passing through the open sector g in order to distinguish between three possible types of curvilinear sectors."

# Local dynamics

Local dynamics:

• in the exterior of Julia set
• on the Julia set
• near parabolic fixed point (inside Julia set)

## Near parabolic fixed point

Orbits near parabolic fixed point and inside Julia set

Why analyze f^p not f ?

Forward orbit of f near parabolic fixed point is composite. It consist of 2 motions:

• around fixed point
• toward / away from fixed point

# How to compute parabolic c values

Type of parabolic parameters:

• root points
• cusps
Parabolic points of period 1 component of Mandelbrot set (parameter plane)
n Internal angle (rotation number) t = 1/n The root point c = parabolic parameter Two external angles of parameter rays landing on the root point c (1/(2^n+1); 2/(2^n+1) fixed point ${\displaystyle z_{\alpha }}$  external angles of dynamic rays landing on fixed point ${\displaystyle z_{\alpha }}$
1 1/1 0.25 (0/1 ; 1/1) 0.5 (0/1 = 1/1)
2 1/2 -0.75 (1/3; 2/3) -0.5 (1/3; 2/3)
3 1/3 0.64951905283833*%i-0.125 (1/7; 2/7) 0.43301270189222*%i-0.25 (1/7; 2/7; 3/7)
4 1/4 0.5*%i+0.25 (1/15; 2/15) 0.5*%i (1/15; 2/15; 4/15; 8/15)
5 1/5 0.32858194507446*%i+0.35676274578121 (1/31; 2/31) 0.47552825814758*%i+0.15450849718747 (1/31; 2/31; 4/31; 8/31; 16/31)
6 1/6 0.21650635094611*%i+0.375 (1/63; 2/63) 0.43301270189222*%i+0.25 (1/63; 2/63; 4/63; 8/63; 16/63; 32/63)
7 1/7 0.14718376318856*%i+0.36737513441845 (1/127; 2/127) 0.39091574123401*%i+0.31174490092937 (1/127; 2/127, 4/127; 8/127; 16/127; 32/127, 64/127)
8 1/8 0.10355339059327*%i+0.35355339059327 0.35355339059327*%i+0.35355339059327
9 1/9 0.075191866590218*%i+0.33961017714276 0.32139380484327*%i+0.38302222155949
10 1/10 0.056128497072448*%i+0.32725424859374 0.29389262614624*%i+0.40450849718747

For internal angle n/p parabolic period p cycle consist of one z-point with multiplicity p[42] and multiplier = 1.0 . This point z is equal to fixed point ${\displaystyle z_{alfa}}$

## Period 1

One can easily compute boundary point c

${\displaystyle c=c_{x}+c_{y}*i}$

of period 1 hyperbolic component (main cardioid) for given internal angle (rotation number) t using this cpp code by Wolf Jung[43]

 t *= (2*PI); // from turns to radians
cx = 0.5*cos(t) - 0.25*cos(2*t);
cy = 0.5*sin(t) - 0.25*sin(2*t);


or this Maxima CAS code:


/* conformal map from circle to cardioid ( boundary
of period 1 component of Mandelbrot set */
F(w):=w/2-w*w/4;

/*
circle D={w:abs(w)=1 } where w=l(t,r)
t is angle in turns ; 1 turn = 360 degree = 2*Pi radians
r is a radius
*/
ToCircle(t,r):=r*%e^(%i*t*2*%pi);

GiveC(angle,radius):=
(
[w],
/* point of  unit circle   w:l(internalAngle,internalRadius); */
w:ToCircle(angle,radius),  /* point of circle */
float(rectform(F(w)))    /* point on boundary of period 1 component of Mandelbrot set */
)$compile(all)$

/* ---------- global constants & var ---------------------------*/
Numerator: 1;
DenominatorMax: 10;
InternalRadius: 1;

/* --------- main -------------- */
for Denominator:1 thru DenominatorMax step 1 do
(
InternalAngle: Numerator/Denominator,
c: GiveC(InternalAngle,InternalRadius),
display(Denominator),
display(c),
/* compute fixed point */
alfa:float(rectform((1-sqrt(1-4*c))/2)), /* alfa fixed point */
display(alfa)
)$ ## Period 2 // cpp code by W Jung http://www.mndynamics.com t *= (2*PI); // from turns to radians cx = 0.25*cos(t) - 1.0; cy = 0.25*sin(t);  ## Periods 1-6 /* batch file for Maxima CAS computing bifurcation points for period 1-6 Formulae for cycles in the Mandelbrot set II Stephenson, John; Ridgway, Douglas T. Physica A, Volume 190, Issue 1-2, p. 104-116. */ kill(all); remvalue(all); start:elapsed_run_time (); /* ------------ functions ----------------------*/ /* exponential for of complex number with angle in turns */ /* "exponential form prevents allroots from working", code by Robert P. Munafo */ GivePoint(Radius,t):=rectform(ev(Radius*%e^(%i*t*2*%pi), numer))$ /* gives point of unit circle for angle t in turns */

GiveCirclePoint(t):=rectform(ev(%e^(%i*t*2*%pi), numer))$/* gives point of unit circle for angle t in turns Radius = 1 */ /* gives a list of iMax points of unit circle */ GiveCirclePoints(iMax):=block( [circle_angles,CirclePoints], CirclePoints:[], circle_angles:makelist(i/iMax,i,0,iMax), for t in circle_angles do CirclePoints:cons(GivePoint(1,t),CirclePoints), return(CirclePoints) /* multipliers */ )$

/* http://commons.wikimedia.org/wiki/File:Mandelbrot_set_Components.jpg
Boundary equation  b_n(c,P)=0
defines relations between hyperbolic components and unit circle for given period n ,
allows computation of exact coordinates of hyperbolic componenets.

b_n(w,c), is boundary polynomial (implicit function of 2 variables).

*/

GiveBoundaryEq(P,n):=
block(
if n=1 then return(c + P^2 - P),
if n=2 then return(- c + P - 1),
if n=3 then return(c^3 + 2*c^2 - (P-1)*c + (P-1)^2),
if n=4 then return(c^6 + 3*c^5 + (P+3)* c^4 + (P+3)* c^3 - (P+2)*(P-1)*c^2 - (P-1)^3),
if n=5 then return(c^15 + 8*c^14 + 28*c^13 + (P + 60)*c^12 + (7*P + 94)*c^11 +
(3*P^2 + 20*P + 116)*c^10 + (11*P^2 + 33*P + 114)*c^9 + (6*P^2 + 40*P + 94)*c^8 +
(2*P^3 - 20*P^2 + 37*P + 69)*c^7 + (3*P - 11)*(3*P^2 - 3*P - 4)*c^6 + (P - 1)*(3*P^3 + 20*P^2 - 33*P - 26)*c^5 +
(3*P^2 + 27*P + 14)*(P - 1)^2*c^4 - (6*P + 5)*(P - 1)^3*c^3 + (P + 2)*(P - 1)^4*c^2 - c*(P - 1)^5  + (P - 1)^6),
if n=6 then return (c^27+
13*c^26+
78*c^25+
(293 - P)*c^24+
(792 - 10*P)*c^23+
(1672 - 41*P)*c^22+
(2892 - 84*P - 4*P^2)*c^21+
(4219 - 60*P - 30*P^2)*c^20+
(5313 + 155*P - 80*P^2)*c^19+
(5892 + 642*P - 57*P^2 + 4*P^3)*c^18+
(5843 + 1347*P + 195*P^2 + 22*P^3)*c^17+
(5258 + 2036*P + 734*P^2 + 22*P^3)*c^16+
(4346 + 2455*P + 1441*P^2 - 112*P^3 + 6*P^4)*c^15 +
(3310 + 2522*P + 1941*P^2 - 441*P^3 + 20*P^4)*c^14 +
(2331 + 2272*P + 1881*P^2 - 853*P^3 - 15*P^4)*c^13 +
(1525 + 1842*P + 1344*P^2 - 1157*P^3 - 124*P^4 - 6*P^5)*c^12 +
(927 + 1385*P + 570*P^2 - 1143*P^3 - 189*P^4 - 14*P^5)*c^11 +
(536 + 923*P - 126*P^2 - 774*P^3 - 186*P^4 + 11*P^5)*c^10 +
(298 + 834*P + 367*P^2 + 45*P^3 - 4*P^4 + 4*P^5)*(1-P)*c^9 +
(155 + 445*P - 148*P^2 - 109*P^3 + 103*P^4 + 2*P^5)*(1-P)*c^8 +
2*(38 + 142*P - 37*P^2 - 62*P^3 + 17*P^4)*(1-P)^2*c^7 +
(35 + 166*P + 18*P^2 - 75*P^3 - 4*P^4)*((1-P)^3)*c^6 +
(17 + 94*P + 62*P^2 + 2*P^3)*((1-P)^4)*c^5 +
(7 + 34*P + 8*P^2)*((1-P)^5)*c^4 +
(3 + 10*P + P^2)*((1-P)^6)*c^3 +
(1 + P)*((1-P)^7)*c^2 +
-c*((1-P)^8) + (1-P)^9)
)$/* gives a list of points c on boundaries on all components for give period */ GiveBoundaryPoints(period,Circle_Points):=block( [Boundary,P,eq,roots], Boundary:[], for m in Circle_Points do (/* map from reference plane to parameter plane */ P:m/2^period, eq:GiveBoundaryEq(P,period), /* Boundary equation b_n(c,P)=0 */ roots:bfallroots(%i*eq), roots:map(rhs,roots), for root in roots do Boundary:cons(root,Boundary)), return(Boundary) )$

/* divide llist of roots to 3 sublists = 3  components */
/* ---- boundaries of period 3 components
period:3$Boundary3Left:[]$
Boundary3Up:[]$Boundary3Down:[]$

Radius:1;

for m in CirclePoints do (
P:m/2^period,
eq:GiveBoundaryEq(P,period),
roots:bfallroots(%i*eq),
roots:map(rhs,roots),
for root in roots do
(
if realpart(root)<-1  then Boundary3Left:cons(root,Boundary3Left),
if (realpart(root)>-1 and imagpart(root)>0.5)
then Boundary3Up:cons(root,Boundary3Up),
if (realpart(root)>-1 and imagpart(root)<0.5)
then Boundary3Down:cons(root,Boundary3Down)

)

)$--------- */ /* gives a list of parabolic points for given: period and internal angle */ GiveParabolicPoints(period,t):=block ( [m,ParabolicPoints,P,eq,roots], m: GiveCirclePoint(t), /* root of unit circle, Radius=1, angle t=0 */ ParabolicPoints:[], /* map from reference plane to parameter plane */ P:m/2^period, eq:GiveBoundaryEq(P,period), /* Boundary equation b_n(c,P)=0 */ roots:bfallroots(%i*eq), roots:map(rhs,roots), for root in roots do ParabolicPoints:cons(float(root),ParabolicPoints), return(ParabolicPoints) )$

compile(all)$/* ------------- constant values ----------------------*/ fpprec:16; /* ------------unit circle on a w-plane -----------------------------------------*/ a:GiveParabolicPoints(6,1/3); a$



## period d

/*
gcc c.c -lm -Wall
./a.out
Root point between period 1 component and period 987 component  = c = 0.2500101310666710+0.0000000644946597
Internal angle (c) = 1/987
Internal radius (c) = 1.0000000000000000

*/

#include <stdio.h>
#include <math.h>		// M_PI; needs -lm also
#include <complex.h>

/*
c functions using complex type numbers
computes c from  component  of Mandelbrot set */
complex double Give_c( int Period,  int n, int d , double InternalRadius )
{
complex double c;
complex double w;  // point of reference plane  where image of the component is a unit disk
// alfa = ax +ay*i = (1-sqrt(d))/2 ; // result
double t; // InternalAngleInTurns

t  = (double) n/d;
t = t * M_PI * 2.0; // from turns to radians

w = InternalRadius*cexp(I*t); // map to the unit disk

switch ( Period ) // of component
{
case 1: // main cardioid = only one period 1 component
c = w/2 - w*w/4; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_1
break;
case 2: // only one period 2 component
c = (w-4)/4 ; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_2
break;
// period > 2
default:
printf("higher periods : to do, use newton method \n");
printf("for each q = Period of the Child component  there are 2^(q-1) roots \n");
c = 10000.0; // bad value

break; }

return c;
}

void PrintAndDescribe_c( int period,  int n, int d , double InternalRadius ){

complex double c = Give_c(period, n, d, InternalRadius);

printf("Root point between period %d component and period %d component  = c = %.16f%+.16f*I\t",period, d, creal(c), cimag(c));
printf("Internal angle (c) = %d/%d\n",n, d);
//printf("Internal radius (c) = %.16f\n",InternalRadius);

}

/*
https://stackoverflow.com/questions/19738919/gcd-function-for-c
The GCD function uses Euclid's Algorithm.
It computes A mod B, then swaps A and B with an XOR swap.
*/

int gcd(int a, int b)
{
int temp;
while (b != 0)
{
temp = a % b;

a = b;
b = temp;
}
return a;
}

int main (){

int period = 1;
double InternalRadius = 1.0;
// internal angle in turns as a ratio = p/q
int n = 1;
int d = 987;

// n/d = local angle in turns

for (n = 1; n < d; ++n ){
if (gcd(n,d)==1 )// irreducible fraction
{ PrintAndDescribe_c(period, n,d,InternalRadius); }
}

return 0;

}


# How to draw parabolic Julia set

All points of interior of filled Julia set tend to one periodic orbit (or fixed point). This point is in Julia set and is weakly attracting.[44] One can analyse only behevior near parabolic fixed point. It can be done using critical orbits.

There are two cases here: easy and hard.

If the Julia set near parabolic fixed point is like n-th arm star (not twisted) then one can simply check argument of zn, relative to the fixed point. See for example z+z^5. This is an easy case.

In the hard case Julia set is twisted around fixed.

## Estimation from exterior

### Long iteration or big step method

Description

• Long iteration method[45]
• The big steps algorithm [46]

Steps

• choose fuction ${\displaystyle f}$ which has fixed parabolic point at origin ( z=0)
• choose internal angle ${\displaystyle \theta ={\frac {m}{k}}}$
• compute ${\displaystyle F_{k}}$  which is an aproximation of higher iterates of function ${\displaystyle f}$  for z close to zero using power series centered at zero ( Taylor series = Maclaurin series )
• find how many terms of power series to use and on which annuli to use specific ${\displaystyle F_{k}}$  experimentally
${\displaystyle F_{k}\approx f^{2^{k}}}$



each ${\displaystyle F_{k}}$  will be used on an annulus

${\displaystyle {\frac {1}{2^{k+K+1}}}<|z|<{\frac {1}{2^{k+K}}}}$

where K is fixed

#### example

Lambda form of complex quadratic polynomial which has an indifferent fixed point with multiplier ${\displaystyle \lambda }$  at the origin[47]

${\displaystyle f_{\lambda }(z)=\lambda z+z^{2}}$

where:

• multiplier of fixed point ${\displaystyle \lambda =e^{2\pi \theta i}}$
• internal angle is an rational number and proper fraction ${\displaystyle \theta ={\frac {m}{k}}}$

Choose

• ${\displaystyle \theta ={\frac {1}{7}}}$  so ${\displaystyle \lambda =e^{(2\pi i)/7}}$
• 30 terms of power series
• aproximated function for annuli k=4,5,...,10 and default function f^n for larger values of z ( outside annuli)
• delta for function equal to 10^-5

(* code by Professor: Mark McClure from https://marksmath.org/classes/Spring2019ComplexDynamics/ *)
n = 7;
f[z_] = Exp[2 Pi*I/n] z + z^2;

Remove[F];
F[0][z_]    = N[Normal[Series[f[z], {z, 0, 30}]]];
Do[F[0][z_] = Chop[N[Normal[Series[F[0][f[z]], {z, 0, 30}]]], 10^-5],  {n - 1}];
Do[F[k][z_] = Chop[N[Normal[Series[F[k - 1][F[k - 1][z]], {z, 0, 30}]]], 10^-5], {k, 1, 10}]

(* define and compile function FF *)
FF = With[{
n = n,
f4 = Function[z, Evaluate[F[4][z]]],
f5 = Function[z, Evaluate[F[5][z]]],
f6 = Function[z, Evaluate[F[6][z]]],
f7 = Function[z, Evaluate[F[7][z]]],
f8 = Function[z, Evaluate[F[8][z]]],
f9 = Function[z, Evaluate[F[9][z]]],
f10 = Function[z, Evaluate[F[10][z]]]
},
Compile[{{z, _Complex}},
Which[
Abs[z] > 1/2^3,
Nest[Function[zz, N[Exp[2 Pi*I/n]] zz + zz^2], z, n],
Abs[z] <= 1/2^9, f10[z],
Abs[z] <= 1/2^8, f9[z],
Abs[z] <= 1/2^7, f8[z],
Abs[z] <= 1/2^6, f7[z],
Abs[z] <= 1/2^5, f6[z],
Abs[z] <= 1/2^4, f5[z],
Abs[z] <= 1/2^3, f4[z],
True, 0]]];

(* iterate 1000 times and then see what happens *)
iterate = With[{FF = FF, n = n},
Compile[{{z0, _Complex}},
Module[{z, i},
z = z0;
i = 0;
While[1/2^9 < Abs[z] <= 2 && i++ < 1000 n,
z = FF[z]];
z],
RuntimeOptions -> "Speed", CompilationTarget -> "C"]];

(* now compute some iteration data *)
data = Monitor[
Table[iterate[x + I*y], {y, Im[center] + 1.2, Im[center], -0.0025},
{x, Re[center] - 1.2, Re[center] + 1.2, 0.0025}],
y];

(* use some symmetry to cut computation time in half *)
center = First[Select[z /. NSolve[f[z] == 0, z], Im[#] < 0 &]]/2 (* center = -0.311745 - 0.390916*I *)
data = Join[data, Reverse[Rest[Reverse /@ data]]];

(* plot it  *)
kernel = {
{1, 1, 1},
{1, -8, 1},
{1, 1, 1}
};

(*
classifyArg = Compile[
{{z, _Complex}, {z0, _Complex}, {v, _Complex}, {n, _Integer}},
Module[{check, check2},
check = n (Arg[(z0 - z)/v] + Pi)/(2 Pi);
check2 = Ceiling[check];
If[check == check2, 0, check2]]];

classified = Map[classify, data, {2}];

convolvedData = ListConvolve[kernel, classified];

ArrayPlot[Sign[Abs[convolvedData]]]


Mathematical Functions of Wolfram language (  :

• Series[f,{x,x0,n}] generates a power series expansion for f about the point ${\displaystyle x=x_{0}}$  to order ${\displaystyle (x-x_{0})^{n}}$ , where n is an explicit integer
• N[expr] gives the numerical value of expr
• Chop[expr,delta] replaces numbers smaller in absolute magnitude than delta by 0
• Normal[expr] converts expr to a normal expression from a variety of special forms.
• Do[expr,n] evaluates expr n times
• Do[expr,{i,imin,imax}] evaluates expr and starts with i=imin
• With[{x=x0,y=y0,…},expr] specifies that all occurrences of the symbols x, y, … in expr should be replaced by x0, y0, ….
• Compile[{{x1,t1},…},expr] assumes that xi is of a type that matches ti.
• Which[test1,value1,test2,value2,…] evaluates each of the testi in turn, returning the value of the valuei corresponding to the first one that yields True
• Nest[f,expr,n] gives an expression with f applied n times to expr.

### Dynamic rays

Parabolic Julia set for internal angle 1 over 15 - made with use of external rays as a aproximation of Julia set near alfa fixed point

One can use periodic dynamic rays landing on parabolic fixed point to find narrow parts of exterior.

Let's check how many backward iterations needs point on periodic ray with external radius = 4 to reach distance 0.003 from parabolic fixed point:

period Inverse iterations time
1 340 0m0.021s
2 55 573 0m5.517s
3 8 084 815 13m13.800s
4 1 059 839 105 1724m28.990s

One can use only argument of point z of external rays and its distance to alfa fixed point (see code from image). It works for periods up to 15 (maybe more ...).

## Estimation from interior

Julia set is a boundary of filled-in Julia set Kc.

• find points of interior of Kc
• find boundary of interior of Kc using edge detection

If components of interior are lying very close to each other then find components using:[48]

color = LastIteration % period


For parabolic components between parent and child component:[49]

periodOfChild = denominator*periodOfParent
color = iLastIteration % periodOfChild


where denominator is a denominator of internal angle of parent comonent of Mandelbrot set.

### Angle

"if the iterate zn of tends to a fixed parabolic point, then the initial seed z0 is classified according to the argument of zn−z0, the classification being provided by the flower theorem" (Mark McClure[50])

### Attraction time

Various types of dynamics

Interior of filled Julia set consist of components. All comonents are preperiodic, some of them are periodic (immediate basin of attraction).

In other words:

• one iteration moves z to another component (and whole component to another component)
• all point of components have the same attraction time (number of iteration needed to reach target set around attractor)

It is possible to use it to color components. Because in the parabolic case the attractor is weak (weakly attracting) it needs a lot of iterations for some points to reach it.

 // i = number of iteration
// iPeriodChild = period of child component of Mandelbrot set ( parabolic c value is a root point between parant and child component
/* distance from z to Alpha  */
Zxt=Zx-dAlfaX;
Zyt=Zy-dAlfaY;
d2=Zxt*Zxt +Zyt*Zyt;
// interior: check if fall into internal target set (circle around alfa fixed point)
if (d2<dMaxDistance2Alfa2) return  iColorsOfInterior[i % iPeriodChild];


Here are some example values:

 iWidth  = 1001 // width of image in pixels
PixelWidth  = 0.003996
AR  = 0.003996 // Radius around attractor
denominator  = 1 ; Cx  = 0.250000000000000; Cy  = 0.000000000000000 ax  = 0.500000000000000; ay  = 0.000000000000000
denominator  = 2 ; Cx  = -0.750000000000000; Cy  = 0.000000000000000 ax  = -0.500000000000000; ay  = 0.000000000000000
denominator  = 3 ; Cx  = -0.125000000000000; Cy  = 0.649519052838329 ax  = -0.250000000000000; ay  = 0.433012701892219
denominator  = 4 ; Cx  = 0.250000000000000; Cy  = 0.500000000000000 ax  = 0.000000000000000; ay  = 0.500000000000000
denominator  = 5 ; Cx  = 0.356762745781211; Cy  = 0.328581945074458 ax  = 0.154508497187474; ay  = 0.475528258147577
denominator  = 6 ; Cx  = 0.375000000000000; Cy  = 0.216506350946110 ax  = 0.250000000000000; ay  = 0.433012701892219

denominator  = 1 ;   i =               243.000000
denominator  = 2 ;   i =            31 171.000000
denominator  = 3 ;   i =         3 400 099.000000
denominator  = 4 ;   i =       333 293 206.000000
denominator  = 5 ;   i =    29 519 565 177.000000
denominator  = 6 ;   i = 2 384 557 783 634.000000


where:

C = Cx + Cy*i
a = ax + ay*i // fixed point alpha
i // number of iterations after which critical point z=0.0 reaches disc around fixed point alpha with radius AR
denominator of internal angle (in turns)
internal angle = 1/denominator


Note that attraction time i is proportional to denominator.

Attraction time for various denominators

Now you see what means weakly attracting.

One can:

• use brutal force method (Attracting radius < pixelSize; iteration Max big enough to let all points from interior reach target set; long time or fast computer)
• find better method (:-)) if time is to long for you

### Trap

Trap = target set

## Estimation from interior and exterior

Julia set is a common boundary of filled-in Julia set and basin of attraction of infinity.

• find points of interior/components of Kc
• find escaping points
• find boundary points using Sobel filter

It works for denominator up to 4.

## Inverse iteration of repelling points

Inverse iteration of alfa fixed point. It works good only for cuting point (where external rays land). Other points still are not hitten.

# Gallery

External examples: