Fractals/Iterations in the complex plane/Parameter plane
Parameter plane
- structure
- algorithms
- models
Structure of parameter plane
[edit | edit source]The phase space of a quadratic map is called its parameter plane. Here:
- is constant
- 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.
Parts of parameter plane
- components ( incuding islands)
- curves
- points
curves or paths
[edit | edit source]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
Paths:
- external rays
- internal rays
- escape route
real slice
[edit | edit source]" each period-doubling cascade superstable orbit ... generate the corresponding chaotic band and the Misiurewicz point that separates the chaotic bands and "[3]
where:
- B is a chaotic region from -2 to MF
- = Feigenbaum point = the Myrberg-Feigenbaum
- A is a periodic region : from MF to 1/4
- is the precedes symbol. The binary relationship: "x precedes y" is written: . It is used to distinguish other orders from total orders.
chaotic region
[edit | edit source]
where:
- is chaotic band
See also
- separators between bands
- It is a part of Sharkovsky ordering
periodic region
[edit | edit source]- 2^n = the powers of 2 in decreasing order, ending with 2^0 = 1."[4]
named parts of Mandelbrot set
[edit | edit source]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 = limiting 1/n limb of main cardioid as n tends to infinity. See demo 2 page 10 from program mandel by Wolf Jung
-
Gap between the head and the body = seahorse valley
-
Double-spirals on the left, seahorses on the right
-
Fragment of Mandelbrot set called the elephant valley
Mini Mandelbrot sets
[edit | edit source]Names:
- midget
- mini mandelbrot set
- baby Mandelbrots
- island = island mu-molecule = island mu-unit [10]
- primitive component
- small copies of itself
- babies Mandelbrot sets = BMSs[11]
Parameter rays of mini Mandelbrot sets [12]
" 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 [13]
See also
- How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is not accessible from main cardioid ( M0) by a finite number of boundary crossing ?
- distorted islands
Julia island or Embedded Julia Set or virtual Julia Sets or medallions
[edit | edit source]- "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[14]
- "This location is called Julia island since it looks like a Julia set but it's actually inside Mandelbrot." [15]
Medallions:
- carrot = non-spiral medallion (-1.6898799090349986e-01 1.0423707254693195e+00 3.0218142193747843e-07 ) type: long double
- cauliflower = single spiral medallion (-0.1543869 1.0308295 3.0218142193747843e-07 ) type: long double
- double spiral medallion (-0.16092059 1.03663239 0.000001 ) type: long double
- triple spiral medallion ( -1.5403777941777627e-01 1.0369221371305641e+00 6.5186720162412668e-07 ) type: long double
hyperbolic componenets
[edit | edit source]order of hyperbolic componenets
[edit | edit source]- Myrberg 1963
- Sharkovsky 1964
- Metropolis 1973
- internal adress by Lau and Schleicher 1994
- rotation number by Devaney 1997
- R2 naming system by R Munafo 1999[16]
- orbit portrait by Milnor 2000
- shrubs by M Romero et al 2004 [17]
shape
[edit | edit source]- ON THE SHAPES OF ELEMENTARY DOMAINS OR WHY MANDELBROT SET IS MADE FROM ALMOST IDEAL CIRCLES by V. Dolotin A.morozov
- Algebraic Geometry of Discrete Dynamics. The case of one variable by V.Dolotin, A.Morozov
Parts of parameter plane
[edit | edit source]- with respect to the Mandelbrot set
- with respect to the wakes
- inside a p/q wake
- outside any wake (???)
Parts of Mandelbrot set according to M Romera et al.:[18]
- 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 of parameter plane:
- wake / limb
- migdet / island / mu-molecule[19]
- hyperbolic component of Mandelbrot set
Wake is a part of parameter plane between 2 parameter rays landing on the same parabolic point ( root point, bond)
Limb is a part of Mandelbrot set between 2 parameter rays landing on the same parabolic point ( root point, bond)
Strict ( math ) definition from the paper Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account by John W. Milnor
Theorem 1.2. The Wake . The two corresponding parameter rays land at a single point of the parameter plane. These rays, together with their landing point, cut the plane into two open subsets
- and
with the following property: A quadratic map has a repelling orbit with portrait if and only if , and has a parabolic orbit with portrait if and only if .
Definitions:
- the Mandelbrot set
- This open set will be called the -wake in parameter space (compare Atela [A]),
- will be called the root point of this wake
- The intersection[22] will be called the -limb of the Mandelbrot set
- The open arc consisting of
- all angles of dynamic rays which are contained in the interior of S1,
- or all angles of parameter rays which are contained in , will be called the characteristic arc for the orbit portrait
Nomenclature types (source of the names):
- mu-ency [23]
- Mandelbrot papers and books
How to move on parameter plane ?
[edit | edit source]-
Values of C for each frame evaluates by equation: C=r*cos(a)+i*r*sin(a), where: a=(0..2*Pi), r=0.7885. Thus, parameter С outlines circle with a radius r=0.7885 and a center at origin of the complex plane.
-
zoom and translation towards Feigenbaum point
-
c=0
-
c=1/4
-
c= 1/4 + 0.05
-
c = 1/4 + 0.029
-
c = 1/4 + 0.035
How to choose step of a move ?
[edit | edit source]- 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[24]
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[25]
Compare
- Numerical calculations and rigorous mathematics
- Reliable Computing, Interval Computations
Types
[edit | edit source]- type of the move
- continuous
- discrete = using sequence of points
- type of the curve
- along radial curves :
- along circular curves :
- equipotentials
- boundaries of hyperbolic components
- internal circular curves
- loops [28]
The dynamics of the polynomials of moving along curves is : [29]
- 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
[edit | edit source]-
point c moves along boundary of main cardioid toward c=0.75 ( root point of period 2 component of Mandelbrot set) using a sequence
External ray
- From Cantor to Semi-hyperbolic Parameters along External Rays[30]
- along parameter external ray for angle 9/31 by David Madore : Its external argument is constantly 9/31 and it approaches the bud's root (~ −0.481763 + +0.531657i) exponentially slowly.
- external angle 1/3 by David Madore
- Cantor to Misiurewicz: along the parameter ray of angle 15/56 by Tomoki Kawahira
- Cantor to Parabolic along the parameter ray of angle 1/3: by Tomoki Kawahira
Boundary of the component
Others
- morphing
- poincare_half-plane_metric_for_zoom_animation by Claude Heiland-Allen
- youtube: Julia sets as C pans over the Mandelbrot set by captzimmo
- youtube : Julia sets about the main cardioid x 1.1 with Mandelbrot set by Thomas Fallon
- youtube: Julia Sets Relative to the Mandelbrot Set by Gary Welz
- you tube : Julia Sets of the Quadratic by Gary Welz
- youtube : Julia set morph around the cardioid / central bulb by blimeyspod
- youtube : Julia set morph / fractal animation - Beyond the Cardioid Perimiter by blimeyspod
- youtube: Julia set morph / fractal animation - Beyond a 2nd Order Bulb by blimeyspod
- Fractals: A tour of Julia Sets by corsec
- shadertoy : Julia - Distance by iq
- Evolving Julia Marco_Gilardi
// 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 // https://www.shadertoy.com/view/MllGzB vec2 c = vec2(-0.754, 0.05*(abs(cos(0.1*iTime))+0.8));
Escape route
[edit | edit source]-
escape route 1/2
Escape route 0
[edit | edit source]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
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:
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
-
center = superattracting
-
attracting
-
parabolic
-
repelling
-
target set
-
Internal level sets
-
binary decomposition
-
Full tile = binary decomposition and internal level sets
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 |
Escape route 1/3
[edit | edit source]-
parameter plane with escape route 1/3
-
animation from center to parabolic
-
superattracting = center for period 1
-
parabolic = boundary
-
superattracting = center for period 3
Plane types
[edit | edit source]Criteria for plane classifications
- fractal formula ( function): c-plane, lambda-plane
- plane transformations
There are many different types of the parameter plane[31] [32]
- by function
- plain c-plane:
- plain lambda plane where
- by transformations [33][34]
- inverted c-plane = 1/c plane:
- exponential plane ( map) [35][36][37]
- unrolled plain (flatten' the cardiod = unroll ) [38][39] = "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)[40]
-
c-plane
-
inverted c plane = 1/c plane
-
plane
-
Unrolled main cardioid of Mandelbrot set for periods 7-13
-
lambda plane
-
1/lambda plane
See also
Parameter space types by dimensions
[edit | edit source]- 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.
Points
[edit | edit source]- real[41]
How to describe c point ?
[edit | edit source]Numerical description
- c value
- Cartesion description
- real part
- imaginary part
- polar description:
- (external or internal ) angle
- ( external or internal) radius, see stability index
- Cartesion description
Symbolic description
- set relation: Julia set interior / boundary / exterior
How to save parameters of the point?
[edit | edit source]- parameter files: files with saved parameter values
- image files with saved parameters
examples of important points
[edit | edit source]Examples from the parameter plane and Mandelbrot set:
- The northwest external angle is 3/8
- The north external angle is 1/4[42]
- The northeast external angle is 1/8[43]
- The west external angle is 1/2
- The east external angle is 0
- The southwest external angle is 5/8
- The south external angle is 3/4
- The southeast external angle is 7/8,
Point Types
[edit | edit source]point
- pixel of parameter plane
- c parameter of complex quadratic polynomial
- complex number
- point coordinate
Criteria
[edit | edit source]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
- in case of exterior point:
- 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), biaccessible, triaccessible, ....
- position of critical point with relation to the Julia set
- Renormalization
Classification
[edit | edit source]There is no complete classification. The "unclassifed" parameters are uncountably infinite, as are the associated angles.
Simple classification
[edit | edit source]- 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[44]
- 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
[edit | edit source]Classification :[45]
- Boundaries of primitive and satellite hyperbolic components:
- 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 ...
How to choose a point from parameter plane ?
[edit | edit source]- 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)[46]
- computing a point from it's known properties
- 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[50]
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[51]
Curves
[edit | edit source]Curves on the parameter plane
- rays
- Parameter External Ray
- internal rays
- equipotential curves
- boundary
- of whole Mandelbrot set
- of hyperbolic components
Algorithms
[edit | edit source]- general or representation functions
- atom domains
- bof60
- The_Lyapunov_exponent
- True shape
- Discrete Langrangian Descriptors
- combinatorial : tuning
- Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection
Models
[edit | edit source]-
Topological model of Mandelbrot set( reflects the structure of the object ). Topological model of Mandelbrot set without mini Mandelbrot sets and Misiurewicz points (Cactus model)
-
Shrub model of Mandelbrot set
-
Topological model of Mandelbrot set using Lavaurs algorithm up to period 12
Structure of the Mandelbrot set
Size or area
[edit | edit source]- mandelbrot-area by Kerry Mitchell ( 2001)
- The size of Mandelbrot bulbs by A.C. Fowler Mark J. Mcguinness
How to to determine the ideal number of maximum iterations for an arbitrary zoom level in a Mandelbrot fractal
[edit | edit source]- a-way-to-determine-the-ideal-number-of-maximum-iterations-for-an-arbitrary-zoom
- Automatic Dwell Limit From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.
- stackoverflow question: how-many-iterations-of-the-mandelbrot-set-for-an-accurate-picture-at-a-certain-z
- stackoverflow question: calculate-a-dynamic-iteration-value-when-zooming-into-a-mandelbrot
- Adaptive maxiter depending on the inverse square root of the magnification
See also
[edit | edit source]- Types of Julia sets
- Exact Coordinates by Robert P. Munafo, 2003 Sep 22.
- Rational Coordinates by Robert P. Munafo, 2012 Apr 22.
Rerferences
[edit | edit source]- ↑ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
- ↑ Generalizations of Douady's magic formula by Adam Epstein, Giulio Tiozzo
- ↑ [Pastor97a] : Harmonic structure of one-dimensional quadratic maps by Gerardo Pastor, Miguel Romera, Fausto Montoya Vitini
- ↑ The On-Line Encyclopedia of Integer Sequences : A005408 = The odd numbers: a(n) = 2n+1
- ↑ mandelmap - A detailed map of the Mandelbrot Set, in a beautiful vintage style by Bill Tavis
- ↑ seahorsevalley From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022
- ↑ mandelbrot-locations by woofractal
- ↑ Mandelbrot Buds and Branches by Timothy Chase
- ↑ Map of the Mandelbrot Set. Copyright © 2005-2011 Janet Parke
- ↑ the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018
- ↑ M. Romera, G. Pastor, A. B. Orue, D. Arroyo, F. Montoya, "Coupling Patterns of External Arguments in the Multiple-Spiral Medallions of the Mandelbrot Set", Discrete Dynamics in Nature and Society, vol. 2009, Article ID 135637, 14 pages, 2009. https://doi.org/10.1155/2009/135637
- ↑ Parameter rays of mini mandelbrot sets
- ↑ Devaney In Global Analysis of Dynamical Systems, ed.: H. Broer, B. Krauskopf, G. Vegter. IOP Publishing (2001), 329-338 or Homoclinic Points in Complex Dynamical Systems
- ↑ embedded julia set from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017.
- ↑ https://www.flickr.com/photos/nonnameavailable/28654921940/
- ↑ https://www.mrob.com/pub/muency/r2namingsystem.html
- ↑ M. Romera et al, Int. J. Bifurcation Chaos 13, 2279 (2003). https://doi.org/10.1142/S0218127403007941 Shrubs in the Mandelbrot Set Ordering
- ↑ SHRUBS IN THE MANDELBROT SET ORDERING by M Romero, G Pastor, G Alvarez, F Montoya
- ↑ mumolecule From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020.
- ↑ fractalforums.com : how-distorted-can-a-minibrot-be
- ↑ distribution From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020
- ↑ Intersection_(set_theory) in wikipedia
- ↑ Mu-Ency - The Encyclopedia of the Mandelbrot Set by R Munafo
- ↑ fractalforums.org : julia-sets-true-shape-and-escape-time
- ↑ fractalforums.org : julia-sets-true-shape-and-escape-time
- ↑ From Cantor to Semi-hyperbolic Parameter along External Rays by Yi-Chiuan Chen and Tomoki Kawahira
- ↑ From Hyperbolic to Parabolic Parameters along Internal Rays by Yi-Chiuan Chen and Tomoki Kawahira
- ↑ math.stackexchange question: parameter-plane-dynamics-of-fixed-points-and-their-preimages-for-standard-quadra
- ↑ Reading escaping trees from Hubbard trees in Sn by Matthieu Arfeux
- ↑ From Cantor to Semi-hyperbolic Parameter along External Rays March 2018Transactions of the American Mathematical Society 372(11) DOI: 10.1090/tran/7839 Yi-Chiuan ChenTomoki KawahiraTomoki Kawahira
- ↑ Alternate Parameter Planes by David E. Joyce
- ↑ exponentialmap by Robert Munafo
- ↑ Twisted Mandelbrot Sets by Eric C. Hill
- ↑ On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I by Luna Lomonaco, Carsten Lunde Petersen
- ↑ mu-ency : exponential map by R Munafo
- ↑ Exponential mapping and OpenMP by Claude Heiland-Allen
- ↑ exponential_mapping_with_kalles_fraktaler by Claude Heiland-Allen
- ↑ Linas Vepstas : Self Similar?
- ↑ the flattened cardioid of a Mandelbrot by Tom Rathborne
- ↑ Stretching cusps by Claude Heiland-Allen
- ↑ Totally real points in the Mandelbrot Set by Xavier Buff, Sarah Koch 2022
- ↑ North by Robert P. Munafo, 2010 Sep 20. From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022.
- ↑ Northeast by Robert P. Munafo, 2010 Sep 20. From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022.
- ↑ From Cantor to Semi-hyperbolic Parameter along External Rays March 2018Transactions of the American Mathematical Society 372(11) DOI: 10.1090/tran/7839
- ↑ stackexchange : classification-of-points-in-the-mandelbrot-set
- ↑ fractalforums : parameter-adjustment-art-or-luck ?
- ↑ interesting c points by Owen Maresh
- ↑ Visual Guide To Patterns In The Mandelbrot Set by Miqel
- ↑ fractalforums : deep-zooming-to-interesting-areas
- ↑ fractalforums.org : parameter-adjustment-art-or-luck
- ↑ fractalforums.org: julia-sets-true-shape-and-escape-time