• December 03, 2021, 08:27:26 AM

Author Topic:  Julia and Mandelbrot sets w or w/o Lyapunov sequences  (Read 10298 times)

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Offline gerrit

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #135 on: September 09, 2021, 12:46:46 AM »
A cardioid so close to the point c_0, then I guess my "high-precision" was way too low. Starting from that cardioid and c_0 is then already closer than what my number type holds and I assume orbits mingle.

But I think Misiurewicz points are off the table altogether:
  • the period-3 cycle equation only has roots with non-vanishing imaginary part (IA subdivision)
  • for period 4, it has been proven that no purely rational (imaginary=0) cycles exist (Morton P, "Arithmetic properties of periodic points of quadratic maps, II", 1998)

And hyperbolic centers are ruled out by (I completely forgot about that):
  • if a (complex) rational n-cycle for n >= 3 exists, it is never super-attracting (Theorem 5, "Arithmetic progressions in cycles of quadratic polynomials", Erkama T.)

That leaves: interior or indifferent of some kind and - my favourite :) - a ghost component.
I first read it in the Erkama article (conjecture after Theorem 1, periodi >= 4).

Thanks for the Rational Zeros theorem. I am working on a subdivision-based method to detect rational zeros and this could help there.
For M-set it seems trivial as the n-th order Mandelbrot polynomial (in c) has leading and constant coefficient of 1 so +- 1 are the only possible rational zeros. (Factor out c of course.)

Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #136 on: October 21, 2021, 11:54:55 AM »
An interesting preliminary result for the multicritical map (4 cps)
\[
f(z) := c\cdot z^2\cdot (1-z^3)
 \]
looking at the c-parameter plane for the critical point cp3=0.7368062997280774 with two peculiarities for me:

1. I don't recall having encountered a period-2 cardioid (yellow) before. Does anyone have another function that shows this?

2. Red are period-1 components. There is only one solution to the equation f(cp3,c)=cp3 to allow for a super-attracting fix-point (c = 2.262014680495755), but what about the separated red bulb at the lower left? It seems to be its own component - with no hyperbolic center? (Or the other myriad of seemingly distinct red components). Maybe period-1 is all "borrowed" from the critical point 0, where the entire complex plane as a c-parameter space allows a superattracting fix-point at z=0.

Image details:
  • periods 1,2,4,6 in red, yellow, green, brownish; escaping, black; gray, bounded
  • double precision, 50,000 maxit, non-rounding controlled, point-sampled, backward period detection
  • region in [-2..0] + i*[-3..-1]

Possible things to do next:
  • high precision (does the period-2 part prevail?)
  • multiplier map (do the components all have a low-multiplier close to 0 center)
  • analytical solutions for hc's of period 2 (does the bulb get a center?)
  • component boundary equation (definite proof of existence of attracting periods)
  • analyzing further critical points

Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #137 on: October 23, 2021, 01:24:24 PM »
Continuing the analysis of the period-2 bulb:

1. high precision orbit construction (left image, upscaled 2-times, a 2x2 pixel grid represents one complex number)

Period-2 pixels from the previous post were re-analyzed with fixed-precision 1764 . 572 decimal digits.

  • purple: period-2 detected at 50,000 iterations, then backward cycle detection with epsilon=10^-25 for  number "equality".
  • yellow: period-2 detected in forward Okazaki-like fashion: repeatedly performing 400 iterations  followed by backward cycle detection
  • a few pixels are judged as period 4 (green, lower component bottom) or 6 (blue, upper component left).  This is probably due to the drawback of forward orbit detection ("warm up") as discussed here: forum link

I looked at some of the orbits to ensure they are not actually fix-points, e.g.
Code: [Select]
c= -1.32+i*-2.28

2-cycle:
  re1=-0.428619696675816230772051672617020621956515898740390752191807858583793404885755269179261922032080526794973215270826556319013053720861187028394335728232787188020603601699495195074620062659994119503123900080912527905736294500824115837245057947984527688795956167088152327737442177015640201713106927939913291888083974602012617717498633536781905191342321679433355127754943731587681605435110604615638115445447415919038403067691011331114774652141254332160216127530058818628797399449670402711591209664509393012014354271083197595282812624106959189275500514071562977947579530629870994738
  im1=-0.740887835775019180819815079722714902725112562143317581585106610087924165414390455867842186456035270604949304959351418200012810950701600686560660081898631573029331913502425134252284250387705409246297969350823032480725618027385892954380509277779335186639682231919312615150126291059485850068553374829548986085912594918192067406291810676252165743129029418780118020140021156064022583549229991094314561045402963617409002298056936135927380718427136029623520216095827516595709787949219816612527865163346914067575535709431021469395719199342091068905828844595771514507601158113746149682
->
  re2=0.719788501447896339879634924387821488555460492569770956438424866718203230703298145289836083273764254598565620003718793087545401110144516837534074498707927278933681477012262731174840506460018334706290560885674069860852481156252435675352544349157465506221153646971831504561994646074216430718013178385912588720763654801433448607169502652819275837415512386652210541286760422195508954960513214036087808319294759912837961535796380096425835770842610607858875377882356409611867128481775306531829519730091865223879486552699340778675494716850390278747050282767224194172393495044729033811
  im2=0.001062863484543068892965777642623265736160900003114170474992356437648189252402909313640379593330229025283288362573685597214999557450293831265346067188784474038905092507375774975703140390864444122631521180273725128401374226820824060904247038424341786783251312071901158702175454987116034840814277737984975290781361557386534155385787079308065832051682996812023331988582330362927447646426508862784941415100799910048752462577568276344786719993190924969476054626327333752379995745299791637792477181355383085992928579108108976931371677929927096350649540675958968336220640281948532722


2. Multiplier map.
The multiplier map for the 2-cycle from item 1) was heatmap colored. The colder the color, the higher the  multiplier (less attracting). The image shows two clear spots of hot colors, consistent with two hyperbolic centers.


So far, results are supporting the existence of a period-2 cardioid directly adjacent to a period-2 bulb.

Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #138 on: October 26, 2021, 01:26:31 PM »
The strict period-2 component equation revealed a factor that describes the presumed period-2 cardioid: If for a complex c (point or interval) there is a complex solution m within the unit disc, the seed belongs to a period-2 component of cardioid-like shape:

\[
0 = m^4-100\cdot m^3+(972\cdot c^3+3750)\cdot m^2+((-729\cdot c^6)+10800\cdot c^3-62500)\cdot m+2916\cdot c^6-67500\cdot c^3+390625
 \]
Direction c -> m
Squares of width 10^-7 were analyzed as complex interval seeds c using reliable interval subdivision to detect existence of a suitable root m.

First image: a hybrid picture, a yellow pixel indicates that such a square seed is fully allowing an attracting 2-cycle and proves the component has positive area. The image was obtained using a near-boundary scanning method repeatedly analyzing pixels with a yellow and an analyzed-but-not-yellow neighbour. Initial pixels were analyzed on the diagonal and then to the right.

CAVEAT:
  • A pixel's width itself is larger at 10^-3, hence the yellow pixels do not form a full covering. A computational proof that the area is simply connected is currently not tractable.
  • definite non-existence of a root was not a sought-for outcome here (a subdivision depth limit was imposed). In theory, the black and  gray parts could convert to yellow at any further subdivision level and a deduction on the shape of the  component would be speculative here.

Direction m -> c
The m,c-equation was algebraically solvable for c and m was ranging over the unit disc in a full area covering manner (2nd image, left). Three cardioid-like shapes occured at symmetric positions. Colors indicate the solution equation that produces that c value, with special color purple (boundary) and white (multiple solutions, attributed to IA overestimation).

For the left image, the statement holds: For each component, the union of the colored pixels form a superset of the component. The purple boundary forms a superset of the component's boundary.

In the right image, the lower cardioid shape was computed in the same manner, but this time m ranging only over a set of squares that cover the unit circle's boudnary. As those c's form a superset of the component's boundary by IA properties, the cusp is indeed a very deep indentation and notably different from any smaller indentation that might lie on the rest of the boundary, no matter how overestimated the marked pixels are.


I am now convinced that this period-2 component is indeed a cardioid.
« Last Edit: October 29, 2021, 02:15:31 PM by marcm200, Reason: formula 0 »

Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #139 on: October 29, 2021, 02:24:58 PM »
Looking at the c-parameter planes when one of the three non-zero critical points is bounded and the others are free in their behaviour showed the existence of two period-2 cardioids (yellow). The image below shows one critical point, the other two depict a similar image with the positions of the yellow cardioids permuted.

Interesting is period-1. There are two qualitatively different regions: The period-1 cardioid (bright red) at the third cardioid position with an attracting fix-point (non-zero), harbouring a hyperbolic center and showing a fractal boundary as to be expected from a minibrot's cardioid.

Secondly, there are the pale red regions, where the chosen critical orbit is  attracted to the origin. Those regions do not have a hyperbolic center and in part do not show a fractal boundary (lower left, the round balls). Do those regions have a special name? (I seem to have overlooked this reading the literature).

The origin plays a prominent role here. It is a critical point, so a critical orbit attracted by a different critical point, and the point z=0 collapses the iteration formula to the constant-0 polynomial, hence the Julia/Fatou theory no longer applies (degree >= 2 as a condition in the theorems).

This very innocently looking polynomial c*z^2*(1-z^3) shows a very feature-rich dynamics.

Offline pauldelbrot

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #140 on: October 29, 2021, 05:32:07 PM »
Zero is a fixed, parameter-independent superattractor here. In a sense every point in your image is a hyperbolic center for it.

Perturb the formula by adding a "+ a" to the end and you'll have a 2nd parameter, and in the plane of that parameter using 0 as the iterated critical point you will find a Mandelbrot figure with a period-1 cardioid whose hyperbolic center is at the origin (i.e., at a = 0). Images in the c-plane with nonzero a should also have a Mandelbrot figure for critical point 0, but in the limit of a -> 0 it will become huge, and at a = 0 infinitely large, so the point at the cardioid's center swallows all of space. There will be extra bubbles or dendrites or other shapes accompanying these Mandelbrot figures, due to the influence of the other critical points and the existence, at some parameter values, of additional finite attractors.

Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #141 on: October 30, 2021, 01:18:16 PM »
The component equation for exact period-1 showed two main factors:

\[
0 = (c^4\cdot m)~\cdot~(m^4-20\cdot m^3+150\cdot m^2+27\cdot c^3\cdot m-500\cdot m-54\cdot c^3+625)
 \]
The first factor describes the whole c-plane being superattracting as pauldelbrot mentioned, the 2nd describes the three cardioids.

Solving explicitly for c and letting m range over the unit circle gave the image below (interval arithmetics using the kv library). But interestingly, every solution described one entire cardioid as opposed to the piece-wise phenomenon for the period-2 cardioids, where each of the 6 solutions describes part of every cardioid.

I presume the period-1 and period-2 cardioids occupy the same c-region (although I have no direct proof).

Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #142 on: November 05, 2021, 01:46:35 PM »
An interesting collection of principal shapes: tricorn, minibrot and logistic map all in the c-parameter-space of one modified logistic formula f(z):

\[
f(z) := |\text{Re}(~g(z)~)|~+~i\cdot |\text{Im}(~g(z)~)| \\
~\\
g(z) := c\cdot (z+t)\cdot (1-(z+t))~-~t \\
t := -\frac{3}{4}-\frac{3}{4}\cdot i
 \]
which in the abs()-free version is a simple translation of the logitisc map, but with the absolute value, the other shapes appear.

The one shape missing is the burning ship. Does anyone have a formula, maybe here one of the tricorns has been replaced by the burning ship, so one could see all the principal quadratic shapes at the same magnification?

The minibrot in the lower left half appears to be perfectly symmetric and completely undistorted. I wonder, if this is a perfect copy of the quadratic non-abs() Mandelbrot set with all relative sizes and positions kept unchanged, especially the period-2 bulb (yellow) being a circle. Could this be proven?

I'm currently attempting to get a component-2 equation, but I have my doubts that this would factorize into simple terms, each describing a single period-2 component.

Image: region [-11 .. +5] x [-11 .. +5] at 25,000 maxit, backward period detection; critical point 5/4+3/4*i, colored by period; gray lines indicate coordinate axis.

Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #143 on: November 21, 2021, 06:16:59 PM »
"Pacman in NY"

Experimenting with exponent functions that map complex numbers to positive integers. The image below shows the c-parameter space for the orbit of the origin (I do not know whether it has a special meaning there, the goal here was to find interesting shapes). Here I used the Manhattan distance to the origin.

\[
z_{new} := z^{1+ \text{ceil}(~\text{g}(z)~) } + c\\
g(z): C \rightarrow N: z \rightarrow 2\cdot(~|\text{Re}(z)|+|\text{Im}(z)|~)
 \]

  • left image: escape time, 2-square: black, escaping; colors, periods 1-7 (see below); gray, bounded but uncyclic; white, cyclic with period larger than 7; escape 2^10, maxit 50,000, backward period detection
  • right image: exponent used if an orbit point lands in that region of the 2-square
  • colors for exponents and periods 1-7, larger: red, yellow, blue, green, cyan, brown, purple, white

My favourite part is the Pacman-like yellow period-2 on the negative real axis.

The function may be regarded as a region-wise defined polynomial with sharp but non-steep exponent changes. It would be interesting to see what happens if one uses a less regular exponent mapping g(z).

Offline Alef

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #144 on: November 21, 2021, 08:17:25 PM »
An interesting collection of principal shapes: tricorn, minibrot and logistic map all in the c-parameter-space of one modified logistic formula f(z):


The one shape missing is the burning ship. Does anyone have a formula, maybe here one of the tricorns has been replaced by the burning ship, so one could see all the principal quadratic shapes at the same magnification?
Burning ship very likely at it's core is a combination of mandelbrot and tricorn. Say all the abs based fractals could be generated by "if else" statement and change of sign (conj function or mirrored mandelbrot)
https://fractalforums.org/fractal-mathematics-and-new-theories/28/mandelweird-3d-burning-ship/4290/msg29210#msg29210

Interesting found of Deseptor
https://fractalforums.org/index.php?action=gallery;sa=view;id=5823

Quote
Description: I discovered recently that the Burning Ship fractal is just 2 Mandelbrots and 2 Mandelbars connected. I whipped up a formula in Ultra Fractal that helps illustrate this. Compare it to a Burning Ship. See the resemblance?


Alsou there were a Fractint's formula by Pauldebrot saying that logistic map escape time fractal could be 2 connected mandelbrot sets. This alsou could be derived shape. Arithmetic mean between mandelbrot set and logistic map:
Quote
Newsgroups: sci.fractals
Date: 1994-12-19 15:17:58 PST

  This formula for Fractint produces weird "doublebrots."
  P2 sets bailout (use a number like 10000 and use floating point or you may
  get garbled results and holes in your set).
  P1 causes the Mandelbrot set to "morph". P1=0 will draw the normal M-set;
  P1=1 will draw the M-set from Fractint's Mandellambda type (derived from
  the verhulst population-dynamic equation). Other values produce all sorts
  of weird things; two floating M-sets, or two fused M-sets, etc. The
  mandellambda turns out to be a fused pair of M-sets.

DoubleBrots    {c=pixel
                pp=1-p1
;adjusted initial value
                z=0.5 :
                a=(z*z+c)
                b=c*z*(1-z)
                z=pp*a+p1*b,
                |z|<=p2}
If you move p1 value you can see how 2 mandelbrot sets joins together forming lambda.

Alsou single iteration of julia set of (0.5,0) and then calculation of Z = Z^@power + C + zJulia will bring 2 mandelbrot sets together creating rotated lambda, like
Code: [Select]
IF iter < @juliaiter
zJulia = zJulia^@power+@Seed

ELSE
Z = Z^@power + C + zJulia

ENDIF
https://fractalforums.org/share-a-fractal/22/some-old-stuff-maybe-fractal-nostalgia/4213/msg30484#msg30484 

So there could be just 2 principal complex quadratic shapes: mandelbrot and tricorn. This sounds as discussion on how many angels can fit on the tip of the needle, anyway.

Colour method (Mega)Discrete Lagrangian Descriptors. 1 one escapes slightly differently, 2 last are identical:
« Last Edit: November 22, 2021, 02:08:23 AM by Alef »
by Edgar Malinovsky aka Edgars Malinovskis.

Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #145 on: November 22, 2021, 01:53:36 PM »
After reading about Alef's joining approach, I revived the old arctan selector functions from post #123 to place a burning ship into the upper left region, replacing that tricorn.
\[
z_{new} := |~\text{Re}(~g(z,c)~)~|+i\cdot |~\text{Im}(~g(z,c)~)~| \\
~\\
g(z,c) := u(c)\cdot(~z^2+c+7-1\cdot i~)~+~(~1-u(c)~)\cdot (~c\cdot (z+t)\cdot(1-(z+t))-t~) \\
u(c) := pos(-\text{Re}(c)-3~)\cdot pos(~\text{Im}(c)+3~) \\
pos(w) := \frac{1}{2}\cdot (~1+\frac{2}{\pi}\cdot arctan(K\cdot w)~) \\
K=10^{10}~;~t=-\frac{3}{4}-\frac{3}{4}\cdot i
 \]
The selector u(c) is almost 1 if c lies in the region [-inf..-3] x [-3..+inf] (the upper left tricorn), almost 0 otherwise and selects if the burning ship iteration or the translated logistic map is predominantly performed (with some perturbation due to the "almost" part). The same selection was undertaken for the critical point.

Plugging all into one closed formula (maxima) gave (c=d+e*i, z iterated orbit point):
Code: [Select]
g: (((2*arctan(1.0E+10*((-d)-3)))/%pi+1)*((2*arctan(1.0E+10*(e+3)))/%pi+1)*(z^2+%i*e+d-%i+7))/4+(1-(((2*arctan(1.0E+10*((-d)-3)))/%pi+1)*((2*arctan(1.0E+10*(e+3)))/%pi+1))/4)*((%i*e+d)*((-z)+(3*%i)/4+7/4)*(z-(3*%i)/4-3/4)+(3*%i)/4+3/4)
cp: ((1-2*((-(3*%i)/4)-3/4))*(1-(((2*arctan(1.0E+10*((-d)-3)))/%pi+1)*((2*arctan(1.0E+10*(e+3)))/%pi+1))/4))/2

The image below was computed using these closed partially expanded but rather lenghty forms directly (after some adjustments to make C++ code). So far I haven't been able to find a simple expression (a polynomial?), let alone one without manual intervention or design.


Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #146 on: November 27, 2021, 02:30:15 PM »
A Julia set of
\[
z_{new} := z^{a(z)}+c
 \]with a(z) delivering the integral exponent to use. The current orbit point z (in the 2-square) is checked in the exponent-to-use image (upper right): if it lands in a yellow region, the exponent returned is 2; 3 for blue; 4 for green.

The Julia set for c=-0.771728515625-0.058837890625*i in the 2-square shows two period-2 cycles (blue, gruen lines; end points are the periodic points) with two intermingled basins (red, yellow). In contrast to a standard polynomial Julia set, the individual basins are not separated by the Julia set's boundary but here lie mingled into one another. Still every bounded point is attracted to a cycle.

Offline marcm200

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #147 on: November 29, 2021, 09:35:40 AM »
Experimenting with some Julia sets for the burning-ship like
\[
z_{new} := |\text{Re}(~z^2+c)~|+i\cdot |\text{Im}(~z^2+c)~|
 \]
For c=0.055419921875-0.815673828125*i an interesting set arose.

A nice combination of nature and fractals (left). interior colored by repeated log of max orbit distance to the origin until the value lies between 0 and 1, then using a heat-map. That revealed a rich substructure, here remarkably resembling dividing cells. The membrane (blue) is almost closing off the cytoplasm (reddish/pink with compartments), and the cell wall (yellow/green) is partially built up.

What surprised me the most was the (possible) existence of a 3-cycle - but all periodic points in one large component (right image, dots and lines). Does the abs() mess things up so strongly so the usual one-basin-surrounded-by-the-Julia-set-boundary is no longer valid?
Code: [Select]
cycle len=3
(0.097415048621100119863,0.032692370025306560777)
(0.063840822514979744806,0.80930437049389147841)
(0.59547799160612480129,0.71234051477040416511) /

I currently am not able to decide
  • is it a numerical error due to my epsilon of number equality, and there actually aren't 3 periodic points
  • are there 3 boundary-enclosed regions that only touch at a point but at this magnification (overview level, 2-square) they appear to be area-connected?
  • or are there actually 3 periodic points in one basin?

I'm currently trying a reliable computing analysis to see what's what.

Offline pauldelbrot

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Re: Julia and Mandelbrot sets w or w/o Lyapunov sequences
« Reply #148 on: November 29, 2021, 04:33:37 PM »
It looks like there are three distinct immediate basins of your periodic points, abutting along fractal curves that meet in a 3-armed spiral. Have you tried using ordinary old convergent smooth iterations on this thing's interior? That might make the immediate basin boundaries clearer.

With a complex-analytic map there would have to be pockets of escaping points along those boundaries, but the use of abs appears to open up the option for there not to be.