Fractals/Iterations in the complex plane/Parameter plane

Structure of parameter planeEdit

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.


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 sliceEdit

" each period-doubling cascade superstable orbit ... generate the corresponding chaotic band   and the Misiurewicz point   that separates the chaotic bands   and   "[3]



  • 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 regionEdit



  •   is chaotic band  

See also

periodic regionEdit

period doubling cascade

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


named parts of Mandelbrot setEdit


  • 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


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

Mini Mandelbrot setsEdit


  • 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]

See also

Julia island = Embedded Julia Set = virtual Julia SetsEdit

  • "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]

hyperbolic componenetsEdit

order of hyperbolic componenetsEdit


Parts of parameter planeEdit

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


  • 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 ?Edit

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 ?Edit

How to choose step of a move ?Edit

  • 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]



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


External ray

Boundary of the component


 // glsl code by iq from
 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
 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 routeEdit

Escape route 0Edit

Escape route for internal angle 0/1


  • 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:


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

Escape route 1/3Edit

Plane typesEdit

Criteria for plane classifications

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

  • by function
    • plain c-plane:  
    • plain lambda plane   where  
  • by transformations [35]
    • inverted c-plane = 1/c plane:  
    • 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 dimensionsEdit

  • 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 ?Edit

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 TypesEdit

point =pixel of parameter plane = c parameter


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


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

Simple classificationEdit

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


  • 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 pointsEdit

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 ...




Structure of the Mandelbrot set

Size or areaEdit

See alsoEdit


  1. Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
  2. Generalizations of Douady's magic formula by Adam Epstein, Giulio Tiozzo
  3. [Pastor97a] : Harmonic structure of one-dimensional quadratic maps by Gerardo Pastor, Miguel Romera, Fausto Montoya Vitini
  4. The On-Line Encyclopedia of Integer Sequences : A005408 = The odd numbers: a(n) = 2n+1
  5. mandelmap - A detailed map of the Mandelbrot Set, in a beautiful vintage style by Bill Tavis
  6. seahorsevalley From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022
  7. mandelbrot-locations by woofractal
  8. Mandelbrot Buds and Branches by Timothy Chase
  9. Map of the Mandelbrot Set. Copyright © 2005-2011 Janet Parke
  10. the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018
  11. Parameter rays of mini mandelbrot sets
  12. 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
  13. embedded julia set from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017.
  16. M. Romera et al, Int. J. Bifurcation Chaos 13, 2279 (2003). Shrubs in the Mandelbrot Set Ordering
  17. SHRUBS IN THE MANDELBROT SET ORDERING by M Romero, G Pastor, G Alvarez, F Montoya
  18. mumolecule From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020.
  19. : how-distorted-can-a-minibrot-be
  20. distribution From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020
  21. Mu-Ency - The Encyclopedia of the Mandelbrot Set by R Munafo
  22. fractalforums : parameter-adjustment-art-or-luck ?
  23. interesting c points by Owen Maresh
  24. Visual Guide To Patterns In The Mandelbrot Set by Miqel
  25. fractalforums : deep-zooming-to-interesting-areas
  26. : parameter-adjustment-art-or-luck
  27. julia-sets-true-shape-and-escape-time
  28. : julia-sets-true-shape-and-escape-time
  29. : julia-sets-true-shape-and-escape-time
  30. math.stackexchange question: parameter-plane-dynamics-of-fixed-points-and-their-preimages-for-standard-quadra
  31. Reading escaping trees from Hubbard trees in Sn by Matthieu Arfeux
  32. 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
  33. Alternate Parameter Planes by David E. Joyce
  34. exponentialmap by Robert Munafo
  35. Twisted Mandelbrot Sets by Eric C. Hill
  36. mu-ency : exponential map by R Munafo
  37. Exponential mapping and OpenMP by Claude Heiland-Allen
  38. exponential_mapping_with_kalles_fraktaler by Claude Heiland-Allen
  39. Linas Vepstas : Self Similar?
  40. the flattened cardioid of a Mandelbrot by Tom Rathborne
  41. Stretching cusps by Claude Heiland-Allen
  42. From Cantor to Semi-hyperbolic Parameter along External Rays March 2018Transactions of the American Mathematical Society 372(11) DOI: 10.1090/tran/7839
  43. stackexchange : classification-of-points-in-the-mandelbrot-set
  44. fractalforums : tricky-mandelbrot-problem