• February 27, 2021, 10:35:48 PM

### Author Topic:  Period doubling in minibrots  (Read 981 times)

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#### marcm200

• 3c
• Posts: 925
##### Period doubling in minibrots
« on: January 15, 2021, 12:12:44 PM »
(From time to time, some questions about the Mandelbrot occur to me, that seem to be very basic - but for which I cannot find the answers myself (or by a (non-exhaustive) search). Maybe someone here knows of some references towards those topics?

1. Area: Due to x-axis symmetry, any hyperbolic component that fully lies in the negative imaginary part of the complex plane has a mirrored twin in the positive part - and those two have obviously the same area.

Aside from those, are there other pairs or even larger groups of components, that have the same area (equal or different periods)?

2. Period doubling: On the real axis, period doubling for the bulb-y components occur up to some seed value. Does period-doubling also occur in the case of minibrots?

If so: Suppose I connect the hyperbolic centers of the minibrot's cardioid with period p and the period-doubled 2p-component with a straight line and extend it outwards. Does then period-doubling follow that line? Or will it be angled?

EDIT: Thread renamed from "Basic questions about the Mandelbrot set" on Feb 4,2021.

« Last Edit: February 04, 2021, 11:37:59 AM by marcm200, Reason: renamed thread title »

#### youhn

• Fractal Friend
• Posts: 14
##### Re: Basic questions about the Mandelbrot set
« Reply #1 on: January 15, 2021, 05:14:46 PM »
Interesting questions.

Some half answers from another noob;

1. All the minibrots not on the x-axis seem to have (more or less) distorted reflectional symmetry and all spokes are also slightly different in exact shape, so strictly speaking I don't think there is absolute reflective symmetry to be found other than around the x-axis (the real numbers). And how to define "area" in this case? The mandelbrot is 1 connected area with only 1 boundary (which can never be found exactly, but the rough shape is very clear for any observer after only a "few" iteration). If you were able to freely add any boundary line in the set, to separate areas, I'm pretty sure it's in theory doable to divide the whole into equally big areas. You would need infinite time to carry out the full work though.
(if my assumptions are incorrect, I'm happy to learn)

2. I have no idea, online orbit visualisations cannot zoom in enough for a simple check. I would guess " yes!".

(no number). No, see answer 1. The line would more or less bend into a curve, depending on the location.

#### marcm200

• 3c
• Posts: 925
##### Re: Basic questions about the Mandelbrot set
« Reply #2 on: January 15, 2021, 06:06:43 PM »
And how to define "area" in this case?

With area I am referring to the area of a specific hyperbolic component only. For periods 1 and 2, the boundaries can be expressed as simple curves and the area is calculable - and for period 3, Giarrusso et. al gave an explicit trigonometric formulation ("A parameterization of the period 3 hyperbolic components of the Mandelbrot set"), so one could in principle integrate to get the area (although all my attempts to do so, failed, but that's a different story).

#### youhn

• Fractal Friend
• Posts: 14
##### Re: Basic questions about the Mandelbrot set
« Reply #3 on: January 16, 2021, 12:41:36 PM »
Uhm, ok. That's a bit outside my comfortable knowledge zone. Though I did find some references. It appears there are 3 methods for p=3 and no known methods for p=4 :

See the summary given on https://en.wikipedia.org/wiki/File:Mandelbrot_set_Components.jpg

• Fractal Frogurt
• Posts: 452
##### Re: Basic questions about the Mandelbrot set
« Reply #4 on: January 16, 2021, 03:02:39 PM »

#### marcm200

• 3c
• Posts: 925
##### Re: Basic questions about the Mandelbrot set
« Reply #5 on: January 30, 2021, 12:05:48 PM »
(no number). No, see answer 1. The line would more or less bend into a curve, depending on the location.

I think I found an example of a non-colinear period-doubling minibrot.

The coordinates

$c_7=0.43237619264199450564-0.22675990443534863039\cdot i \\ c_{14}=0.4325688150887696537-0.22873440581356344059\cdot i \\ c_{28}=0.43266973566541755414-0.22933968667591089763\cdot i$

form a period doubling triple of hyperbolic centers consisting of a cardioid with period 7, a bulb with period 14 and a further out bulb of period 28.

Determining the angle of the two vectors starting from the hyperbolic center of the period-14 bulb towards the other two component centers led to a value of 176.1 degree. Among the other 32 period-doubling minibrots of cardioid period-7 I analyzed (brute-force), most have an angle of 179 or even 180 (in case of the imaginary coordinate being 0).

The image below shows the minibrot (period-7 components in red, period-14 in green and period 28 blue). The centers are connected with a line. The zoom on the right depicts the period-14 bulb indicating the angle of the vectors.

Interestingly, this minibrot is (for my experience) quite asymmetrical, maybe this is a hint where to look for further and more pronounced angular examples. Does anyone have coordinates for other distorted minibrots?

Limitations
• all calculations were done in a point-sampling, numerical, non-rounding controlled manner, so I cannot  rule out that the angle is a numerical error, e.g. due to hyperbolic center misdetection. Their multipliers however are 3*10^-14, 2*10^-9 and 3*10^-12, which seem a valid approximation to the centers. I will perform a rigorous interval-arithmetics analysis with this example and hopefully more to come.
• the components consist solely of points where numerically a cycle could be detected. I think it is  plausible to assume they extend a bit further out and then the cardioid touches the 14-bulb and that the 28-bulb, but I have no formal proof for that and cannot rule out the possibility that there are very tiny, area bearing components in between.

Technical details
• I used the list of definite root regions from (forum link) and computed a small neighbourhood around each, only keeping points where a period-7,14 or 28 cycle could be detected.
• components were separated further (and extraneous single pixel components were discarded) by only keeping pixels whose two layers of neighbours exhibit the same period.
• components were geometrically detected with the hyperbolic center being the pixel with the smallest multiplier. This center pixel was then refined by analyzing a 16x16 grid on the pixel and its neighbour layer to get a better approximation.
• components were automatically judged as cardioids or bulbs using the w-method from (forum link).
• brute-force analysis was done in a 2^12 x 2^12 pixel resolution with constant max iteration of 25,000 and double precision.

#### youhn

• Fractal Friend
• Posts: 14
##### Re: Basic questions about the Mandelbrot set
« Reply #6 on: January 31, 2021, 04:44:20 PM »
Based on some visual exploration of the mandelbrot set using Kalles Fraktaler, I would think the most warped minibrots are the (relative) biggest ones around a (mini/mandelbrot). Once zooming into nearby smaller minibrots, they all tend to become more symmetrical again. Though one could argue that all minibrots are 100% equal, and that it's only the space that get's warped.

They say the mset is fully connected, so shouldn't the calculated areas touch (on a single point)? Attached is a render that approximates the border of the set from the outside, with the above images (manually) projected onto it.

• Fractal Frogurt
• Posts: 452
##### Re: Basic questions about the Mandelbrot set
« Reply #7 on: January 31, 2021, 07:47:14 PM »

#### marcm200

• 3c
• Posts: 925
##### Re: Basic questions about the Mandelbrot set
« Reply #8 on: February 01, 2021, 12:20:21 PM »
They say the mset is fully connected, so shouldn't the calculated areas touch (on a single point)?

The connectedness of the Mset proves that there is a path - between all the marked components - that never leaves the Mset (including its boundary). But as the images are point-sampled, there is always distance between two  numbers for two adjacent pixels whose escape behaviour is not depicted in the image. So there could be a chain of components, just pretending that the interesting components touch.

For now, I would need the equations for the boundaries of those components to verify that they intersect (similar to forum link) - however, those are, afaik, not computed (computable).

---------------------------------
The cosine of $$\angle (c_{14}\rightarrow c_7,c_{14}\rightarrow c_{28})$$ was calculated via interval arithmetics and outward rounding to be (a lot more digits were identical but by far not necessary):

Code: [Select]
[-0.997681155858468473453590591576224796926146179..-0.997681155858468473453590591576224796926146178]
which does neither contain -1 nor 1 and hence the angle is different from 180 (and 0) degree and the three components' center do not form a straight line.

It would be interesting to see, where the next period-doubled bulbs locate. How does the angle between two consecutive components change? Maybe some angle bends in the opposite direction, so that the chain of centers does not deviate too much from a line?

I think they cannot always bend to the same side as then a spiral would form and the Mset ended there..

Algorithmic details
• a small pixel neighbourhood around the numerically found hyperbolic center approximation was computed and the rectangle then subjected to IA subdivision root finding. Initial regions for subdivision were pairwise disjoint.
• a tailor-made high precision fix-point number type was used at 5400 integer decimals and 432 fractionals.
• the center for period p was analyzed with the polynomial describing the hyperbolic centers of period p and all its divisiors. Once a root region B was found at the expected period, the polynomials for the divisors (and their divisors) were tested to verfiy that B does not contain a hyperbolic center of divisor period.
• as a confirmation, for periods 7 and 14, the cell-mapping TSA was used to verify that the final root region (outward rounded to fit double precision) only harbours Mset numbers that exhibit a period p cycle. For period 28, at my current hard- and software limitation at L20, no cycle could be detected.
• as a negative control, the center regions were used at only 5 fractional digits and provided a cosine straddling -1.

The root regions [r0..r1]+i*[i0..i1] below reliably enclose the respective hyperbolic centers at an accuracy of better than 10^-214.

Code: [Select]
// root region c_7:r0 =0.432376192641994507824669648086921373887850639876994036204241251002834166302290895459969880303985995847338524802499444595200139126565289328917522446637247277753980718637680733065714274518276324279285081052783625783121513424012188391558069593824380745958530100454254238044374424786558161055111526716393202087428806028412316685703111009341084146259126312336041409904148286853260661785467832218239446642892087439714838594933819516297044r1 =0.432376192641994507824669648086921373887850639876994036204241251002834166302290895459969880303985995847338524802499444595200139126565289328917522446637247277753980718637680733065714274518276324279285081052783625783121513424012188391558069648428917485674576039785791878850040668940564750026196306054719933688170910324060790346329317130140865968857126358491381615883279107259383922862591976490966012725662050413597983312182379224654431i0=-0.226759904435348618697876559971698972120232191460389969044495193517667854349636330672103012270364858819519874060651583702193268713427265050756625137903812712764755318330898876406456506084088377450876489339901304121596135307743420618839822328177985217816158378759852162298045714727826739096886913258157027118769496626929545737508142300141583121198700990476858360963121728470907468531139252016557772603451162537976660814331862144136271i1=-0.226759904435348618697876559971698972120232191460389969044495193517667854349636330672103012270364858819519874060651583702193268713427265050756625137903812712764755318330898876406456506084088377450876489339901304121596135307743420618839822258566458208371156535664292813797169939879876703259529201843219683736636987920502527659463190369717022309058064190379368206284531554011808440523360133916508583258631679236555497017441223702361499// root region c_14:r0 =0.432568811799922736600099472465418444280795099414752108944054334585526076638071723523287688129803072713971685785956316187407730111564636458980274281505126053906187688758386000412447372294514132443493865457899983686386050916949753506107593999604298200655101640658366651637204928066312141756640378889717076471108698548286598813495831024576945358943372800169544662714158665148836149220156531017677638232777499872570573361622335097447913r1 =0.432568811799922736600099472465418444280795099414752108944054334585526076638071723523287688129803072713971685785956316187407730111564636458980274281505126053906187688758386000412447372294514132443493865457899983686386050916949753506107593999604298200655101640658366651637204928066312141756640378889717076471108698548286598813495831024576945358943372800779146711133940622122276961669135766270437742741644311805768657847986576171336495i0=-0.228734414339756139285265418096148959018123393842058177114661793624735462497185051266270383039206071792914416488180703781037657604227588208595148610615155847887721546278565051713880906906104420106955493267214789196215659070055734556151016312766072935438630867350108119526500824409548776941297019146391234284903304515024231182037023498097923423484740988431831757871966960814682954475921467796447502217656699540255738662017375937785323i1=-0.228734414339756139285265418096148959018123393842058177114661793624735462497185051266270383039206071792914416488180703781037657604227588208595148610615155847887721546278565051713880906906104420106955493267214789196215659070055734556151016312766072935438630867350108119526500824409548776941297019146391234284903304515024231182037023498097923423484740987685550434107796026172344084396557335131124140011627804186747904087068974868526327// root region c_28:r0 =0.432669818784130669539677962288015718057584368534966008335078278147638921849466453209300644500416434899611715441943009051869858029361248696871454534866877623543225372268656684228338700662282974572960913707473852413978391956252438172554546064149824172457766594454427356267484815805755488837322130854030605498865141101363580658296114102512043236813490826220061135922666561256473986569865906720610501785676014870798219943630611090734779r1 =0.432669818784130669539677962288015718057584368534966008335078278147638921849466453209300644500416434899611715441943009051869858029361248696871454534866877623543225372268656684228338700662282974572960913707473852413978473139337829328627509075418224600380736679482038998934418198544590908053059736038792758161280639154211983975036318849897695717533930373772224977764025919042942439457255636396029345632867674903706921544266519580811572i0=-0.229339671447065501400885319925200893099553632310193579062445535224188960174462845810881314954535721712263062955421632200010042794928160912087383920767579040479821377825587723064616206222728195265212927630534782657281923332501488170233368706683818113112125843598537213654856258848813460811313818992217552231606073336971381071066045756676336224608296534439178712742418054472400135114703238164679059867458697420625368367611296280988619i1=-0.229339671447065501400885319925200893099553632310193579062445535224188960174462845810881314954535721712263062955421632200010042794928160912087383920767579040479821377825587723064616206222728195265212927630534782657281824538063636007533208913160748769660019022246491827545586227530730990738006378188114698284822608246592829814630106994369699471619460547041174953475469170650393206335951435865862177808936735285455392087009645858240357

#### marcm200

• 3c
• Posts: 925
##### Re: Basic questions about the Mandelbrot set
« Reply #9 on: February 03, 2021, 01:30:27 PM »
The current smallest angle I found using the hyperbolic center root regions is now: 175.1 degree with a period doubling cascade of 12-24-48 (image 1 below). The overall smallest with 174.4 (cardioid period 64) was posted by claude in the old forum (credits to Adam for pointing the link out, post #7 here).

For the small cycle lengths, computing every minibrot and looking at it was feasible, for periods > 10 however, this turns out to be too much work, especially as the distorted minibrots are quite rare. As I only have a portion for e.g. period 24, I might miss the interesting ones.

I plan on putting an automated filter first, that calculates the eccentricity of the cardioid and then would subject only the most distorted ones to period-doubling.

The first examples show a tendency between smaller angle and larger eccentricity (however not linear and not from an exhaustive comparison):

Code: [Select]
cardioid       angle             period p       p<-2p->4p         eccentricity   cardioid center------------------------------------------------------------------------------------------------------5              177.1             1.35           0.35925922475800742273-0.64251373713854231795*i6              176.5             1.47           0.4433256333996235532-0.37296241666284651872*i7              176.1             1.55           0.43237619264199450564-0.22675990443534863039*i8              175.8               1.52         0.40489966517512215871-0.14582036376658927268*i10             175.4             1.65           0.35681724849231194474-0.069452865466830299157*i12             175.1             1.72           0.32558950955066034982-0.038047880934755723414*i
The eccentricity was calculated on the image itself in a geometrical manner and defined as the ratio of two distances to the cusp (2nd image):
$ecc := \frac{|cusp \rightarrow B|}{|cusp \rightarrow A|}$.

Left is an almost symmetric cardioid of period 12 with an eccentricity of 1.02, the right being the distorted one with 1.72. The two lines touching the lobes of the cardioid are by construction perpendicular to the (extended) line from the hyperbolic center to the cusp.

#### marcm200

• 3c
• Posts: 925
##### Re: Period doubling in minibrots
« Reply #10 on: February 04, 2021, 11:48:21 AM »
(I renamed the topic to reflect its content).

Looking at the locations of the most disturbed minibrots (among those I analyzed, non-exhaustive) gave an interesting pattern (upper image): The cardioid centers (yellow small squares) seem to lie on an arc (yellow polygon) heading towards the main cardioid's cusp.

It would be a great speed-up if I could predict other eccentric minibrots by extending that curve, so I did some number-fitting. After several trial and error attempts, there emerged a nice relationship between the center in spherical coordinates and the period.

Predicted cardioid hyperbolic center c for period p:

$||c|| = 2.6309381689 \cdot p^{-0.8451186203}\\ \\ -atan2(c) = 4045.9267434218 \cdot p^{-2.5745309728}$

where atan2 is the C++ implementation of the arc tangens respecting the quadrant.

Has anyone encountered other curves with distorted minibrots on them?

I will next try to predict some non-computed periods and subject that region to IA subdivision to find the hyperbolic centers in a rigorous way. As I am only looking for some distorted minibrots I do not demand this regression curve to be perfect, it would suffice if it could predict the overall region of eccentric minibrots of some nearby periods (not too far off the regression curve's base data), then I could use those to refine the regression and proceed.

Charts and regression analyis thanks to OpenOffice.

Code: [Select]
period    center of cardioid 5           0.359259224758007-0.642513737138542*i6           0.443325633399624-0.372962416662846*i7           0.432376192641994-0.226759904435349*i8           0.404899665175122-0.145820363766589*i9           0.378608124132929-0.098558011118382*i10          0.356817248492312-0.0694528654668303*i11          0.339410819995598-0.0506682851626427*i12          0.325589509550660-0.0380478809347557*i14          0.305676541495292-0.0229934263740991*i

#### marcm200

• 3c
• Posts: 925
##### Re: Period doubling in minibrots
« Reply #11 on: February 05, 2021, 01:25:51 PM »
Using the prediction from last post, I could find distorted minibrots for periods between 5 and 43, the latter being the current record holder: eccentricity of 1.91 and an angle of 174.34.

Code: [Select]
period     eccentricity   angle           cardioid center                           p<-2p->4p------------------------------------------------------------------------------------------15          1.79            174.8           0.29844800890399547644-0.018383367322073254635*i17          1.80            174.7           0.28756611704687790043-0.012281055409848253349*i21          1.84            174.6           0.27436979919551240936-0.0062585874913430950342*i25          1.87            174.5           0.2652783219046058732 -0.0037120599898783183101*i31          1.90            174.4           0.26095224231422198269-0.0018410978175303128503*i43          1.91            174.34          0.2556052938434967281 -0.00066797029763820438275*i
Prediction algorithm
1. the regression polynomial is computed using data up to period p.
2. The potential location of that last period p is predicted and its Euclidean distance to the known coordinates is used as an error value E.
3. The location of the minibrot cardioid center for the desired period q=p+1, $$c_q$$  is predicted and a box A with c_q as center and side length 2E is subjected to IA subdivision to find the hyperbolic centers of period q.
4. As a heuristics, I usually only subject the intersection between A and the box B formed by the origin and c_p to root finding, assuming the curve keeps its left turn character.

• Fractal Frogurt
• Posts: 452
##### Re: Period doubling in minibrots
« Reply #12 on: February 05, 2021, 03:55:49 PM »
looks cool
These cureves inside components are not internal rays ? ( in main cardioid only some rays are straight lines)

#### marcm200

• 3c
• Posts: 925
##### Re: Period doubling in minibrots
« Reply #13 on: February 06, 2021, 10:29:19 AM »
looks cool
These cureves inside components are not internal rays ? ( in main cardioid only some rays are straight lines)
Thanks. I have not read much about internal/external rays in the Mset, so I cannot give an answer. The straight lines in my last image only connect the 3 hyperbolic centers to get a visual on the relevant angle (green component) and a line connecting the cardioid's hc to its cusp to calculate the eccentricity. If the lines happen to be (part of) internal rays, that's purely coincidental.

• Fractal Frogurt
• Posts: 452
##### Re: Period doubling in minibrots
« Reply #14 on: February 07, 2021, 02:53:13 PM »
Is p the period of the biggest minibrot in the m/n wake ?

If yes then it's period is p = n+1

What is the relation between
• period p
• internal angle m/n of main cardioid

and eccentricity ?

Is eccentricity proportional to p ?

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