# Structure of parameter plane

The phase space of a quadratic map is called its parameter plane. Here:

• ${\displaystyle z0=z_{cr}\,}$  is constant
• ${\displaystyle c\,}$  is variable

There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.

The parameter plane consists of :

• The Mandelbrot set
• The bifurcation locus = boundary of Mandelbrot set
• Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set [1]

Structure of parameter plane from programmers point of view:

• exterior of M-set
• Each component is surrounded by an atom domain ( disc with radius 4 times bigger, for cardioids radius has about the square root of the size).
• Each component has a nucleus at its center, which has a periodic orbit containing 0.

## curves

The Mandelbrot set contains smooth curves:

• the intersection with the real axis M ∩ R = [−2, 1/4] = real slice of Mandelbrot set
• the main cardioid of M, which is the set of parameters c for which fc has an attracting or indifferent fixed point (of course, this is smooth except at the cusp c = 1/4).[2]
• boundary of period 2 hyperbolic componenet, which is a circle

## real slice

" each period-doubling cascade superstable orbit ... generate the corresponding chaotic band ${\displaystyle B_{n}}$  and the Misiurewicz point ${\displaystyle m_{n}}$  that separates the chaotic bands ${\displaystyle B_{n=1}}$  and ${\displaystyle B_{n}}$  "[3]

${\displaystyle B\prec MF\prec A}$

where:

• B is a chaotic region from -2 to MF
• ${\displaystyle c=MF=m_{\infty }=b_{\infty }=-1.4011551890......}$  = Feigenbaum point = the Myrberg-Feigenbaum
• A is a periodic region : from MF to 1/4
• ${\displaystyle \prec }$  is the precedes symbol. The binary relationship: "x precedes y" is written: ${\displaystyle x\prec y}$ . It is used to distinguish other orders from total orders.

### chaotic region

${\displaystyle B_{0}\prec B_{1}\prec B_{2}\prec ...\prec B_{\infty }}$

where:

• ${\displaystyle B_{m}}$  is chaotic band ${\displaystyle B_{m}=(2n+1)\cdot 2^{m}}$
 ${\displaystyle 3*2^{m}\prec 5*2^{m}\prec 7*2^{m}\prec ...\prec \infty }$



### periodic region

• 2^n = the powers of 2 in decreasing order, ending with 2^0 = 1."[4]

${\displaystyle 2^{\infty }\prec ...\prec 2^{2}\prec 2^{1}\prec 2^{0}}$

## named parts of Mandelbrot set

Valley:

• double spiral
• spindle
• seahorse valley/coast ( or Seahorse Valley East ) [5]
• main cardioid seahorse valley = Gap between the head ( period 2 component) and the body (or shoulder = main cardioid). Particularly the upper one part.[6]
• disk 3 seahorse valley = Gap between period 1 and period 3
• Elephant Valley = Gap near cusp of main cardioid. Here the antennae resemble the trunks of elephants[7]
• elephant coast = boundary of main cardioid near cusp
• Scepter Valley = gap between period 2 and period 4 component, Also known as Seahorse Valley West or Sceptre Valley. "Scepter Valley" where double spirals have scepters coming (echoing the spindle) coming off of all their tips.[8] [9]
• double spiral valley =
• double spiral coast = boundary of period 2 component near main cardioid
• triple spiral coast ( valley) - boundary of period 3 componnet near period 1 cardioid

Parts

• Peacock eye = shrub ( decoration) from Seahorse valley = Principal Misiurewicz point with all branches
• seahorse
• elephant

## Mini Mandelbrot sets

Names:

• midget
• mini mandelbrot set
• baby Mandelbrots
• island = island mu-molecule = island mu-unit [10]
• primitive component

Parameter rays of mini Mandelbrot sets [11]

" a saddle-node (parabolic) periodic point in a complex dynamical system often admits homoclinic points, and in the case that these homoclinic points are nondegenerate, this is accompanied by the existence of infinitely many baby Mandelbrot sets converging to the saddle-node parameter value in the corresponding parameter plane." Devaney [12]

## Julia island = Embedded Julia Set = virtual Julia Sets

• "A structure comprised of filaments, resembling a Julia set in appearance, which has a higher delta Hausdorff dimension than filaments in the immediately surrounding region. They are sometimes also called Julia islands or virtual Julia Sets." Robert Munafo[13]
• "This location is called Julia island since it looks like a Julia set but it's actually inside Mandelbrot." [14]

## Parts of parameter plane

Parts of Mandelbrot set according to M Romera et al.:[17]

• main cardioid
• q/p family (= q/p limb)
• periodic part: period doubling cascade of hyperbolic components which ends at the Myrberg-Feigenbaum point
• the Myrberg-Feigenbaum point
• chaotic part: shrub

Not that her q/p not p/q notation is used

Components:

• wake / limb
• migdet / island / mu-molecule[18]
• hyperbolic component of Mandelbrot set

Names source

• mu-ency [21]
• Mandlebrot

# How to choose a point from parameter plane ?

• take point ( and check it's properities)
• taking parameter choosing by other people (visual choose)
• clicking on parameter points and see what you have ( random choose)[22]
• computing a point from it's known properties
• For (parabolic point ) choose hyperbolic component ( period, number) and internal angle (= rotation number) then compute c parameter.
• Misiurewicz points [23]
• morphing
• interesting areas[25]
• zoom

Is parameter tweaking an acquired art or just random chance?

 "From my own experience with monocritical polynomials and Lyapunov diagrams, all my images I found purely by chance.
For z^2+c e.g. as long as you're in the same hyperbolic component, the shape changes only in the sense, that a fat spiral might become thinner, but the number of arms stays constant.
If you move the c value out of that    component into another - and if this 2nd component is not directly attached to the first, then, I'm not aware of a direct way of telling what it would look like there.

Usually I perform a parameter walk just computing black and white escape images with periodicity. Then for  the interesting shapes/periods I apply a color walk with some gradients that looked fine in previous images.
But that's  more or less guessing.

If you want to turn more into the constructing-an-image from a vision you have, you might try two articles: genetic algorithm and Leja points"  marcm200[26]

Mandelbrot set : z^6+ A*z+c How do you find such intereseting examples ?

" I'm running from time to time an A,c-parameter space walk (brute force) in a rather wide grid (-2..+2 in ~0,01 or larger steps) for the family z^n+A*z+c, adding a small random dyadic fraction to the 4d coordinates to get variation.
Following numerically the orbits of the critical points with a rather high max it of 25000 it's possible to get the number of attracting cycles and their length to some accuaracy level in a decent time. If those A,c-parameter pairs pass some filters (mostly sum of length of cycles and diversity) I scan through small overview pictures manually.
Then I use interesting A,c pairs (shape-wise or from the filter values) and some small deviations from it to compute level 10-12 TSA images, as sets wiith similar shapes can show a different dynamical behaviour w.r.t. the level at which interior cells emerge.
I'll take the fastest one and see how many cycles can be detected up to level 18-19." marcm200[27]

# How to move on parameter plane ?

## How to choose step of a move ?

• fixed size step / adaptive size step
• minimal size of the move

"For c values that could not be represented accurately by C++ double data type I calculated the images using interval arithmetics with tiny intervals (border values being fractions of 2^25, interval width 2^-25) encompassing the published value (as needed for real or imaginary part or both) (values were computed using wolfram-alpha to multiply large integers). Given is the left (smaller) border value, the right is obtained by adding one." marcm200[28]

What is the reason for : 2^-25  ?

"When I started with this article back in March this year, my initial formulas were z^2+c and z^2+c*z.

In expanded form with real coordinates: z=(x+i*y) and c=(d+e*i):

( x²-y² + d, 2xy + e )  or ( d*x + x² - e*y - y² , e*x + d*y + 2x*y )

To accurately represent a sum, the widest two terms can be apart is 53 bits (mantissa precision for C++ double) and all other must lie in this range.

The smallest non-zero value of x,y is "axis range / pixel count", i.e. 4 (escape radius of 2, hence axis -2..+2) divided by 2^refinement level.

So for x^2 this goes to 2^-26 as the lowest possible value for x. And since d,e are multiplied with x in the 2nd formula, the same goes for d and e. As I do not like to work "on the edge" I used a buffer of 1-2 bits and came to  the lowest value of 2^-25 for d,e (and refinement level limit of 27 which is currently outside a reasonable range).

For the 1st formula z^2+c, the seed value could go as little as 2^-48 (stated in the article) as it is only added.

For long double and float128 one could go lower in both formulas, but I haven't explored that."marcm200[29]

Compare

## Types

The dynamics of the polynomials of moving along curves is : [31]

• for external ray: ”stretch” the dynamic on the basin of infinity. The argument of φPa,b (2a) stay unchanged ( fixed)
• for equipotential: twist the dynamics in the annulus between the Green level curves of the escaping critical point and of the critical value. The modulus of φPa,b (2a is fixed

## Examples

External ray

Boundary of the component

Others

 // glsl code by iq from https://www.shadertoy.com/view/Mss3R8
float ltime = 0.5-0.5*cos(time*0.12);
vec2 c = vec2( -0.745, 0.186 ) - 0.045*zoom*(1.0-ltime);

 // glsl code by xylifyx  from https://www.shadertoy.com/view/XssXDr
vec2 c = vec2( 0.37+cos(iTime*1.23462673423)*0.04, sin(iTime*1.43472384234)*0.10+0.50);

 // by Marco Gilardi
vec2 c = vec2(-0.754, 0.05*(abs(cos(0.1*iTime))+0.8));


### Escape route

#### Escape route 0

Escape route for internal angle 0/1

Steps

• nucelus of period 1 component ( c = 0) Fixed point alfa is supperattracting fixed point. Julia set is connected.
• along internal ray 0. Imaginary part of parameter c is zero. 0 < cx < 0.25. Fixed point alfa is attracting fixed point. Julia set is connected.
• parabolic point c = 1/4. Fixed point alfa is parabolic fixed point. Julia set is connected.
• along external ray 0. Imaginary part of parameter c is zero. 0.25 < cx. Fixed point alfa is repelling fixed point. Julia set is disconnected

Here parabolic implosion/ explosion ( from connected to disconnected ) occurs. In parabolic point child periodic points coincides with parent period points

evolution of dynamics along escape route 0 ( parabolic implosion)
parameter c location of c Julia set interior type of critical orbit dynamics critical point fixed points stability of alfa
c = 0 center, interior connected exist superattracting atracted to alfa fixed point fixed critical point equal to alfa fixed point, alfa is superattracting, beta is repelling r = 0
0<c<1/4 internal ray 0, interior connected exist attracting atracted to alfa fixed point alfa is attracting, beta is repelling 0 < r < 1.0
c = 1/4 cusp, boundary connected exist parabolic atracted to alfa fixed point alfa fixed point equal to beta fixed point, both are parabolic r = 1
c>1/4 external ray 0, exterior disconnected disappears repelling repelling to infinity both finite fixed points are repelling r > 1

Stability r is absolute value of multiplier at fixed point alfa:

${\displaystyle r=|m(z_{\alpha })|}$


c = 0.0000000000000000+0.0000000000000000*I 	 m(c) = 0.0000000000000000+0.0000000000000000*I 	 r(m) = 0.0000000000000000 	 t(m) = 0.0000000000000000 	period = 1
c = 0.0250000000000000+0.0000000000000000*I 	 m(c) = 0.0513167019494862+0.0000000000000000*I 	 r(m) = 0.0513167019494862 	 t(m) = 0.0000000000000000 	period = 1
c = 0.0500000000000000+0.0000000000000000*I 	 m(c) = 0.1055728090000841+0.0000000000000000*I 	 r(m) = 0.1055728090000841 	 t(m) = 0.0000000000000000 	period = 1
c = 0.0750000000000000+0.0000000000000000*I 	 m(c) = 0.1633399734659244+0.0000000000000000*I 	 r(m) = 0.1633399734659244 	 t(m) = 0.0000000000000000 	period = 1
c = 0.1000000000000000+0.0000000000000000*I 	 m(c) = 0.2254033307585166+0.0000000000000000*I 	 r(m) = 0.2254033307585166 	 t(m) = 0.0000000000000000 	period = 1
c = 0.1250000000000000+0.0000000000000000*I 	 m(c) = 0.2928932188134524+0.0000000000000000*I 	 r(m) = 0.2928932188134524 	 t(m) = 0.0000000000000000 	period = 1
c = 0.1500000000000000+0.0000000000000000*I 	 m(c) = 0.3675444679663241+0.0000000000000000*I 	 r(m) = 0.3675444679663241 	 t(m) = 0.0000000000000000 	period = 1
c = 0.1750000000000000+0.0000000000000000*I 	 m(c) = 0.4522774424948338+0.0000000000000000*I 	 r(m) = 0.4522774424948338 	 t(m) = 0.0000000000000000 	period = 1
c = 0.2000000000000000+0.0000000000000000*I 	 m(c) = 0.5527864045000419+0.0000000000000000*I 	 r(m) = 0.5527864045000419 	 t(m) = 0.0000000000000000 	period = 1
c = 0.2250000000000000+0.0000000000000000*I 	 m(c) = 0.6837722339831620+0.0000000000000000*I 	 r(m) = 0.6837722339831620 	 t(m) = 0.0000000000000000 	period = 1
c = 0.2500000000000000+0.0000000000000000*I 	 m(c) = 0.9999999894632878+0.0000000000000000*I 	 r(m) = 0.9999999894632878 	 t(m) = 0.0000000000000000 	period = 1
c = 0.2750000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.3162277660168377*I 	 r(m) = 1.0488088481701514 	 t(m) = 0.0487455572605341 	period = 1
c = 0.3000000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.4472135954999579*I 	 r(m) = 1.0954451150103321 	 t(m) = 0.0669301182003075 	period = 1
c = 0.3250000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.5477225575051662*I 	 r(m) = 1.1401754250991381 	 t(m) = 0.0797514300099943 	period = 1
c = 0.3500000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.6324555320336760*I 	 r(m) = 1.1832159566199232 	 t(m) = 0.0897542589928440 	period = 1



Real slice of parameter plane: imag(c) = 0
parameter c value description of c locations fixed points Julia set basins target set ( petal)
1/4 < c point c is on the external ray 0 both fixed points are repelling disconnected only ona basin of attraction ( infinity)
c = 1/4 cusp of main carioid both fixed points are parabolic ( belong to Julia set) connected = Cauliflower circle
0 < c < 1/4 inside main cardioid, along internal ray 0 Treść komórki Treść komórki
c = 0 center of main cardioid Treść komórki Treść komórki
0 < c < 3/4 inside main cardioid, along internal ray 1/2 Treść komórki Treść komórki
c = 3/4 root point ( parabolic) Treść komórki Treść komórki
3/4 < c < 1.0 inside period 2 component, along internal ray 0 Treść komórki Treść komórki
c = 1.0 center of period 2 component Treść komórki Treść komórki
1/0 < c < 5/4 inside period 2 component, along internal ray 1/2 Treść komórki Treść komórki
c = 5/4 root point ( parabolic) Treść komórki Treść komórki

stability index of period 1 points period 1 points on dynamic plane period 1 points on parameter plane
changes from attractive through indifferent to repelling moves from interior of Kc to its boundary moves from interior of componetnt of M-set to its boundary

# Plane types

Criteria for plane classifications

There are many different types of the parameter plane[33][34]

• by function
• plain c-plane: ${\displaystyle z=z^{2}+c}$
• plain lambda plane ${\displaystyle z=\lambda z(1-z)}$  where ${\displaystyle c={\frac {\lambda }{2}}-{\frac {\lambda ^{2}}{4}}}$
• by transformations [35]
• inverted c-plane = 1/c plane: ${\displaystyle z=z^{2}+c^{-1}}$
• exponential plane ( map) [36][37][38]
• unrolled plain (flatten' the cardiod = unroll ) [39][40] = "A region along the cardioid is continuously blown up and stretched out, so that the respective segment of the cardioid becomes a line segment. .." ( Figure 4.22 on pages 204-205 of The Science Of Fractal Images)[41]

## Parameter space types by dimensions

• 1D ( 1 real parameter):
• 2D ( 1 complex parameter): standard Mandelbrot set, here space is a 2D plane
• 4D ( 2 complex parameters) : the family f(z) = z^n+A*z+c by marcm200
• 6D ( 3 complex parameters) : six dimensional space of the complex parameters m, b, and d used in the formula f(x)=mx(1-x)(x+b)/(x+d) by Valannorton

One can only show 2D slice in the multi dimensional space.

# How to describe c point ?

Numerical description

• c value
• Cartesion description
• real part
• imaginary part
• polar description:
• (external or internal ) angle
• ( external or internal) radius, see stability index

Symbolic description

• set realtion: Julia set interior / boundary / exterior

# Point Types

point =pixel of parameter plane = c parameter

## Criteria

Criteria for classification of parameter plane points :

• arithmetic properties of internal angle (rotational number) or external angle
• in case of exterior point:
• type of angle : rational, irrational, ....
• preperiod and period of angle under doubling map
• in case of boundary point :
• preperiod and period of external angle under doubling map
• preperiod and period of internal angle under doubling map
• set properties ( relation with the Mandelbrot set and wakes)
• interior
• boundary
• exterior
• inside wake, subwake
• outside all the wakes, belonging to a parameter ray landing at a Siegel or Cremer parameter,
• geometric properties
• number of external rays that land on the boundary point : tips ( 1 ray), biaccesible, triaccesible, ....
• position of critical point with relation to the Julia set
• Renormalization

## Classification

There is no complete classification. The "unclassifed" parameters are uncountably infinite, as are the associated angles.

### Simple classification

• exterior of Mandelbrot set
• Mandelbrot set
• boundary of Mandelbrot set
• interior of Mandelbrot set ( hyperbolic parameter)
• centers,
• other internal points ( points of internal rays )

Definitions

• a parameter c in the boundary of Mandelbrot set ∂M is semi-hyperbolic if the critical point is non-recurrent and belongs to the Julia set[42]
• A typical example of semi-hyperbolic parameter is a Misiurewicz point: We say a parameter ˆc is Misiurewicz if the critical point of fcˆ is a pre-periodic point.

### partial classification of boundary points

Classification :[43]

• Boundaries of primitive and satellite hyperbolic components:
• Parabolic (including 1/4 and primitive roots which are landing points for 2 parameter rays with rational external angles = biaccesible ).
• Siegel ( a unique parameter ray landing with irrational external angle)
• Cremer ( a unique parameter ray landing with irrational external angle)
• Boundary of M without boundaries of hyperbolic components:
• non-renormalizable (Misiurewicz with rational external angle and other).
• renormalizable
• finitely renormalizable (Misiurewicz and other).
• infinitely renormalizble (Feigenbaum and other). Angle in turns of external rays landing on the Feigenbaum point are irrational numbers
• non hyperbolic components ( we believe they do not exist but we cannot prove it ) Boundaries of non-hyperbolic components would be infinitely renormalizable as well.

Here "other" has not a complete description. The polynomial may have a locally connected Julia set or not, the critical point may be rcurrent or not, the number of branches at branch points may be bounded or not ...

Examples:

# Models

Structure of the Mandelbrot set