• October 20, 2021, 03:47:17 AM

### Author Topic:  Accumulation point of period-doubling  (Read 364 times)

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#### marcm200

• 3e
• Posts: 1072
##### Accumulation point of period-doubling
« on: October 02, 2021, 12:46:15 PM »
I'm currently collecting methods to (reliably) compute the accumulation point of the period-doubling cascade in the real-valued x^2+c mapping. Before delving into the literature, however, I wantred to come up with my own algorithm based on subdivision and interval arithmetics.

The goal is to construct two sequences of reals that enclose the accumulation point, W, in an interval form  $$[~A_k~..~B_k~]$$ and in the end use this as a target point for a pi-route.

The right sequence B_k is fairly straightforward: One uses the coordinates of the hyperbolic centers in the  cascade, c_k.

The basic idea is to compute all real-valued solutions to the equation for hyperbolic centers up to period 2^N  (which also gives the centers of divisor periods, so the entire start of the cascade), order them on the real axis and take the rightmost N+1 solutions.

Results for centers up to 2^12 are (took about 2 hrs hours):
Code: [Select]
  period 2^12 [-1.4011551704528392292559146881103515625..-1.401155170425190590322017669677734375]  period 2^11 [-1.401155102023321887827478349208831787109375..-1.4011551020224578678607940673828125]  period 2^10 [-1.4011547826253809034824371337890625..-1.401154782404191792011260986328125]  period 2^9  [-1.4011532911472022533416748046875..-1.40115329026244580745697021484375]  period 2^8  [-1.40114633165299892425537109375..-1.401146324574947357177734375]  period 2^7  [-1.40111385047435760498046875..-1.401113793849945068359375]  period 2^6  [-1.4009622669219970703125..-1.400961360931396484375]  period 2^5  [-1.400256500244140625..-1.4002492523193359375]  period 2^4  [-1.3969659423828125..-1.396907958984375]  period 2^3  [-1.38171630859375..-1.381484375]  period 2^2  [-1.3109765625..-1.30912109375]  period 2^1  [-1.0178125..-0.988125]  period 2^0  [-0.0975..+0.14]
Optimizations
Technically I am not calculating all real solutions. I keep a decreasingly sorted list of disjoint intervals (touching at the boundary is allowed) that either harbour a unique root or are currently unjudgeable. Then I repeatedly analyze the rightmost N+1 of those intervals, subdivide them and judge for having a unique root or definitely none. The other intervals are simply copied.

If, at some point, the rightmost N+1 intervals harbour all a unique root, the algorithm is finished successfully and all relevant centers have been detected.

Technical detals
• existence and uniqueness pf roots are classified using the univariate interval Newton operator  (theorem 5, N Kamath, "Subdivision algorithms for complex root isolation: empirical comparisons").
• the starting region that encloses definitely the cascade is set to [-1.76 .. 0.14]. The right end  being positive, so the main cardioid's center will be found, the left end is to the more negative of the cusp of the period-3 minibrot on the real axis  for complex z^2+c, which analytically is exactly at -1.75 and outside the cascade.
• start interval points and width are chosen so that none of the interval boundaries will be a  dyadic fraction as long as precision is sufficient. Therefore all centers lie inside an interval and not at the border values. Period 2^0 and 2^1 will be found rather quickly, and all longer periods have an irrational center (thanks to gerrit for the observation in forum link).
• interval number type with precision 684 . 216 decimal digits currently sufficient.

The left sequence
Next I turn to the left sequence based on my (half-proven) current working hypothesis that the rest of  the cascade (the part to the left of a bifuraction point of the 2^N-period bulb in the complex) is properly contained in:
$[~c_N-3\cdot(c_{(N-2)}-c_N)~..~c_N~]$

#### claude

• 3f
• Posts: 2033
##### Re: Accumulation point of period-doubling
« Reply #1 on: October 02, 2021, 04:26:59 PM »
There is a sequence of hyperbolic centers at periods 3, 6, 12, .. 3×2^n  tending rightwards towards the accumulation point from the left, being the period 3 real island tuned by each next component in the period doubling cascade.  I conjecture they are the rightmost real centers of those periods, and I conjecture that their limit is the accumulation point.

• Fractal Frankfurter
• Posts: 573
##### Re: Accumulation point of period-doubling
« Reply #2 on: October 02, 2021, 06:51:20 PM »

#### marcm200

• 3e
• Posts: 1072
##### Re: Accumulation point of period-doubling
« Reply #3 on: October 03, 2021, 10:12:08 AM »
There is a sequence of hyperbolic centers at periods 3, 6, 12, .. 3×2^n  tending rightwards towards the accumulation point from the left, being the period 3 real island tuned by each next component in the period doubling cascade.  I conjecture they are the rightmost real centers of those periods, and I conjecture that their limit is the accumulation point.
That's interesting: Two doubling cascades starting from the period-3 minibrot - one to the left inside the minbrot itself with unique periods and one to the right on non-adjacent components (probably multiple ones with the same period?).

@Adam: Your skript shows a numerical value for the accumulation point. Do you know how this was determined?

----------

For a fully algorithmic sequence (a_k), here I used the centers of periods   2*(odd number > 1)  .  Starting in the region [-1.76 .. -1.26] then all roots to the respective hc equation lie outside the doubling cascade and any one of them could be used as (a_k).

The rightmost so far found with reliable computing
Code: [Select]
least negative center found of period 94 or divisor (!= 1, != 2):  [-1.43035763245396 ..-1.43035763245333}
which -. together with the c_12 from the first post, gave one accurate digit of W

$\text{W}~=~ 1.4\text{___}$
(not much, but it comes with a mathematical guarantee and I think is acceptable for one day's effort)

The drawback: I do not know whether this type of component centers comes arbitrarily close to the accumulation point. Maybe they stall at some other number? But as long as I get identical leading digits between (a_k) and (b_k), W will be enclosed tighter.

• Fractal Frankfurter
• Posts: 573
##### Re: Accumulation point of period-doubling
« Reply #4 on: October 03, 2021, 03:41:12 PM »
@Adam: Your skript shows a numerical value for the accumulation point. Do you know how this was determined?

This is not direct answer, but all what I have found

Decimal expansion of the accumulation point of the logistic map

There are many papers about precise computations of Feigenbaum constants

https://github.com/sjmeijer/feigenbaum

https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/1over2_family

HTH
« Last Edit: October 04, 2021, 09:01:12 PM by Adam Majewski, Reason: https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/1over2_family »

#### marcm200

• 3e
• Posts: 1072
##### Re: Accumulation point of period-doubling
« Reply #5 on: October 04, 2021, 11:36:23 AM »

I looked for hc's with exact periods not a power of 2 (divisor periods were ruled out) to approach the accumulation point from the left. The following contains many periods of claude's 3*2^n form. I'm quite positive I could have detected/selected the missing powers in between as well.

The accumulation point's accurate digits are now (still purely algorithmically found)

$\text{W}~=~-1.4011551\text{___}$

based on
Code: [Select]
   exact period      hyperbolic center------------------------------------------------------------------------------  10 240              [-1.40115519835214694170606628551117467308128405691517936298290397410383858502191944950254764989949762821197509765625..-1.4011551983518575684973718162319931597284299499917163932747029021753069899514088092473684810101985931396484375]*  6 144 = 3*2^11     [-1.4011552040606122296218617559248885848342513360691827966654499082466145409853197634220123291015625..-1.401155204056101201409444834259703128342185029852230089054444805984900312978425063192844390869140625]*  3 072 = 3*2^10     [-1.4011552590049358568599676378837486582018610585531614987075954559259116649627685546875..-1.401155258924249518178484183121304859985688175301465907551801137742586433887481689453125]*  1 536 = 3*2^9      [-1.401155515587492863977353782455026985631629798945141374133527278900146484375..-1.4011555150492335101673273480788546770103852878719408181495964527130126953125]     640              [-1.401159594516876036357673485088781717422534711658954620361328125..-1.4011595863814136260039160120083323590733925811946392059326171875]*    192 = 3*2^6      [-1.4011885079502824336117328130058012902736663818359375..-1.401188329753202044258841851842589676380157470703125]*     48 = 3*2^4      [-1.4018850803375244140625..-1.40187057971954345703125]      60              [-1.40742431640625..-1.407423095703125]
and the 2^13 period-doubling component at
Code: [Select]
c_13: [-1.401155185099705704487860202789306640625..-1.4011551850962496246211230754852294921875]

#### marcm200

• 3e
• Posts: 1072
##### Re: Accumulation point of period-doubling
« Reply #6 on: October 06, 2021, 05:33:50 PM »
Attempting the enclosement center subdivision this time starting from the period-3 hc on the real axis, gave 8 accurate fractional digits for that accumulation point ()assuming there exists one) using the 3*2^12 center of the cascade and a period-9216 (= 3^2*2^10) center to the left:
$W_3~=~-1.77981807\text{___}$
claude's 3*period suggestion seems to work here as well: Using the in-cascade periods 3*2^n and the out-cascade periods 3*(3*2^n) (a few other components approaching from the left had the same periodicity structure).
Code: [Select]
center of exact period 3^2*2^10 (outside cascade)     [-1.779818077031502020659550140021363702288342313606595013197875232435762882232666015625..-1.77981807703041392291081094638958902897328540060328805338940583169460296630859375]center of exact period 3*2^12 (inside)     [-1.77981807551592282834462821483612060546875..-1.7798180755157818566658534109592437744140625]
Is it known what nature this (or any) accumulation point has w.r.t. to number theory? A rational number,  algebraic or transcendental? Or in some subset like Brjuno-numbers?

Regarding characteristics, I'm undecided between a rationally indifferent point (e.g. the cascade travelling into the cusp region of a minibrot on the real axis) or a Misiurewicz point - or something different?

#### lkmitch

• Fractal Phenom
• Posts: 56
##### Re: Accumulation point of period-doubling
« Reply #7 on: October 06, 2021, 05:37:38 PM »
I did some exploration of this years ago using Newton's method. The period of the disk whose center is being sought is known, and its approximate location can be found using an estimate of the Feigenbaum constant (~4.6692) and the last known center. Since the disks get smaller as the period increases, the accumulation point is the limit of the disk centers.

Here's some of what I came up with. I used the UBASIC language to calculate these values. Based on them, the accumulation point is approximately x = -1.40115518909.

Period 262144
x = -1.401155189090251033181770560583444036347193168272471441245261367863241822075001320965423878918807940761305859228751192252131597674252576491746840333939693093730785180509439998407177461884732043     f(x) = 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001994     Approx Feigenbaum constant = 4.669201609074452566227981520370886753946099646679618270214759104174216232218331252115391377450331072883335437422585227573345431005726594884368803676788279203477482728794667534497622208785380761

Period 524288
-1.401155189091665188307196810081654665031802961459167811540487696793757830527120425853540101556147626116893933766771059134322925978268976962942754662953366672786737113473009338909783794873017929     0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008829391     4.669201609096878794705135037864783677622666525741836726064298772302704408840101335674395312063888699405841681772460778862792478575742234590320122559262793104969373245311653832559840230430927275

Period 1048576
-1.401155189091968057029478931032870596150319761089860169031702604560140034535609583625572649044685749209252985721291938379365835767943137662498558332948696714341390220891894934332628723842939248    -0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000022112864     4.669201609101681681186960160845801729928088893244076170976791076415820432966366679529922720514475494724034090674755784799531875432740922189370340807244635153783290121618647654581771235818384323

#### marcm200

• 3e
• Posts: 1072
##### Re: Accumulation point of period-doubling
« Reply #8 on: October 07, 2021, 10:03:51 AM »
Period 262144
Period 524288
Period 1048576
Quite high periods there. Are you using an error-controlled calculation procedure (interval, ball, affine, ... )? If so, what libraries? They seem to be very fast (I'm currentlöy attempting 2^15 which might take a while).

#### lkmitch

• Fractal Phenom
• Posts: 56
##### Re: Accumulation point of period-doubling
« Reply #9 on: October 08, 2021, 05:19:41 PM »
Quite high periods there. Are you using an error-controlled calculation procedure (interval, ball, affine, ... )? If so, what libraries? They seem to be very fast (I'm currentlöy attempting 2^15 which might take a while).

No, just brute force and a lot of precision.

#### marcm200

• 3e
• Posts: 1072
##### Re: Accumulation point of period-doubling
« Reply #10 on: October 09, 2021, 11:56:47 AM »
A conditional proof for the enclosement hypothesized in post #1 (based on geometric series, majoranting and interval enlargement):

Let c_N be a hyperbolic center of period 2^N in the real doubling-cascade starting from the origin.  Then the accumulation point, W, is contained in:

$W~\in~[~c_N-3\cdot (c_{(N-2)}-c_N)~..~c_N~]$

So far, the proof (attached pdf), is valid under the condition that the Feigenbaum constant is >= 2  (I have to find an explicit citation for that) and for sufficiently large, but unspecified, N.

Assuming the validity for the current c_14/c_12 hc's, however, there are no additional accurate digits  when compared to approaching from the left with a non-period-doubling component (neither for the period-3 minibrot's cascade).

For now, the theoretical advantage - definite convergence and the need to only subdivide for one hc - does not translate into a practical improvement.

#### marcm200

• 3e
• Posts: 1072
##### Re: Accumulation point of period-doubling
« Reply #11 on: October 10, 2021, 12:30:10 PM »
I used lkmitch's approximation center values c'_N for periods 2^N with N=18,19,20 to validate them with the Newton-Interval-operator. This was fairly easy (took only a few minutes) to get three intervals
$[~L_N~..~R_N~] = [~~c'_N - 10^{-29}~..~c'_N + 10^{-29}~]$
Each of these contain a unique root of exact period 2^N and no divisor periodic centers.

The next step, however, turned out to be intractable with subdivision: Proving that the interval $$[~R_n~..~\frac{1}{4}~]$$ does not contain a center of exact period 2^N (technically instead of 1/4 I only go to the last verified center in the cascade and use its right interval end). After a day's calculation, less than 1% of the interval has been discarded.

If the freeness could be shown, then the found center is the first when coming from the positive real axis and hence the one in the period doubling cascade. For now, it could theoretically be one that is close, but outside.

So next I have to find fast methods to prove an interval does not have a zero for a given polynomial:  Any suggestions/experience?

Two ideas:
• monotonicity and sign change recursively for the polynomial and (all) its derivatives. But this might not be working due to the large degree?
• reliable interior DE to "travel" from the hc outwards a few times to discard part of the interval-in-question as being inside a component's interior. Then, far away from the center, subdivision might work in fairly large intervals?

#### claude

• 3f
• Posts: 2033
##### Re: Accumulation point of period-doubling
« Reply #12 on: October 10, 2021, 06:42:34 PM »
Could you somehow prove by induction that the next component is in the cascade (and not beyond it) by using the asymptotic geometrical scaling that was conjectured by Milnor and proven by Lyubich (afaik).

I think it's true that between a period 2^n component in the cascade and a real period 2^n component (same n) not in the cascade is a component of lower period, possibly of period 3 2^(n-2).

#### marcm200

• 3e
• Posts: 1072
##### Re: Accumulation point of period-doubling
« Reply #13 on: October 11, 2021, 12:53:20 PM »
Could you somehow prove by induction that the next component is in the cascade (and not beyond it) by using the asymptotic geometrical scaling that was conjectured by Milnor and proven by Lyubich (afaik).
If I understand correctly:

It would be perfect could one bound the distance from one hc to the next in the cascade from above as a simple function of the previous distance and ensuring that not the entire rest of the cascade is contained within that distance. Then, once I've found the desired period, all unclear subintervals need no longer be analyzed. And finding a root could be sped up by using a somewhat more random half-depth-half-breadth search.

As the hc-distance ratio converges to the Feigenbaum constant, and if doing so in a geometric manner, I think this should be possible, maybe even something as simple as 1/3*previous distance from some low N onwards?

Thanks, claude.

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