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Author Topic:  Constructing polynomials with attracting cycles  (Read 714 times)

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Offline marcm200

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Constructing polynomials with attracting cycles
« on: March 01, 2020, 06:35:44 PM »
After constructing Julia sets with a predefined shape (forum link), here's another method in that line: Constructing polynomials that have an attracting cycle visiting predefined points in the plane.

Constructing a polynomial with a cycle is a simple task by just interpolating the desired points. However that function rarely has an attracting cycle, at least I haven't found one - but this article has:

Quote
Construction of mappings with attracting cycles
Zhang, W., Agarwal, A. P.
Computers and Mathematics with applications 45 2003 1213-1219
full text: https://www.sciencedirect.com/science/article/pii/S0898122103000889

The authors only talk about real valued polynomials. Nevertheless I used their method to construct complex polynomials and afterwards checked numerically if the resulting formula owns the desired properties (analysing the orbits of the critical points and using Newton's method).

The algorithm uses Hermite interpolation. Its input is a set of periodic points and a set of desired derivatives at those points. That way one can control the final multiplier to be attracting or even super-attracting.

The intrigueing idea is to construct a polynomial that consists of two parts: one that exhibits the periodic behaviour and a 2nd additive term that is zero at the periodic points - but the derivative is correcting the derivative of the periodicity part so that it finally can become attracting.

Here's a first simple example:

Desired cycle: based on the quadratic basilica: -1+0i -> 0+0i -> -1+0i
and the derivatives were randomly set to: 0.5+0i and 0+0.5i so that the multiplier would be below 1.

The final polynomial (as by formula 13 in the article) resulted in:

p(z)=(2.5+0.5i)*z^3+(3.5+1i)*z^2+(0+0.5i)*z^1+(-1+0i)

The Julia set is contained in the 4-square (using Douady's estimate). Periodicity (thin yellow line in the image below) and multiplier were as expected.

Next step now is to construct a longer cycle in an interesting shape and (very) finally writing some words.


Linkback: https://fractalforums.org/index.php?topic=3349.0

Offline marcm200

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Re: Constructing polynomials with attracting cycles
« Reply #1 on: March 02, 2020, 09:39:34 AM »
"The house"

Constructing an attracting period-5 cycle in the shape of a house - and permutating the order in which the periodic points are traversed.

Code: [Select]
# position derivative
1 -0.125-0.125i 0.25+0i
2 -0.125+0.125i 0+0.25i
3 0+0.25 i 0-0.25i
4 0.125+0.125i 0+0.25i
5 0.125-0.125i -0.25+0.25i

The degree-9 polynomial for the order 1->2->3->4->5->1 is:

Code: [Select]
p(z)=(59768.832-385220.608)*z^9+(-153567.232-24854.528i)*z^8+(512+7168i)*z^7+(-448-576i)*z^6+(180.736-1056.384i)*z^5+(-293.936-50.544i)*z^4+(0.5+15i)*z^3+(-1.4375-1.5625i)*z^2+(0.3695-1.66425i)*z^1+(-0.17184375+0.00559375i)

The image below shows some permutations (periodic cycle marked in cyan lines, oppsite colors than usual: black = exterior, white = interior).

The overall shape remained the same (I computed all possible combinations) - except for the lower left one, which I particularly like as it possesses an additional, unexpected fix point (the exaggerated green point between the green intersecting lines).

Offline marcm200

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Re: Constructing polynomials with attracting cycles
« Reply #2 on: March 03, 2020, 01:05:29 PM »
A while ago I asked a question about whether any pair of attracting periodic lengths do occur in a polynomial (forum link).

With the above article and some modifications, the answer is in the affirmative: For any positive integers n,m one can construct a polynomial with superattracting cycles of those lengths and additionally with any given locations in the complex plane.

Below is an example where I put a square (cyan lines) and a triangle (green) into one polynomial (Julia set in the 11-square, cropped out).

Code: [Select]
p(z)=(-185069.98885599966-28779.597150006157i)*z^13+(-320903.2134525415-157919.47351622628i)*z^12+(-315719.3790644852-101990.91826610581i)*z^11+(-276924.36018819432-14896.022791463125i)*z^10+(-163537.04558488802+27255.235582973983i)*z^9+(-69968.277391602838+35371.68830630045i)*z^8+(-17631.574363018117+21441.241058748121i)*z^7+(-1871.7837135907575+9572.4353604984062i)*z^6+(795.77790444995117+1759.8198661876359i)*z^5+(836.36702810849238+258.78509404812041i)*z^4+(106.72887347105063+58.936192381961405i)*z^3+(4.1319822925307754-29.085935503481021i)*z^2+(3.7912920048968379-2.3081860986015941i)*z^1+(-0.42993374393102507-0.069566057852280316i)

Method's details

Given 2 desired cycle lengths n,m and a set of complex points ai with cycles a1..an and an+1..an+m, the following polynomial shows two super-attracting cycles of lengths n and m at those positions:

\[
H(z) := \left [
\sum\limits_{i=1}^{n-1} a_{i+1} \cdot h_i(z)
+ a_1 \cdot h_n(z)
\right ] + \left [
\sum\limits_{i=n+1}^{n+m-1} a_{i+1} \cdot h_i(z)
+ a_{n+1} \cdot h_{n+m}(z)
\right ]
 \]

The first term (in square brackets) exhibits the period n, the 2nd term the cycle of length m.

The function hi(z) is designed to fulfill the following demands (formula 9 in the article):

  • hi(aj) vanishes at every periodic point (independent of the cycle) other then ai, i.e. hi(a j) = 1 iff i=j, else 0
  • the derivative hi'(aj) = 0 at every periodic point, so every cycle is superattracting

\[
h_i(z) := [1-2 \cdot l_i'(a_i) \cdot (z-a_i)] \cdot [l_i(z)]┬▓
 \]

where li is a characteristic function (no demand for the derivative here) for periodic point ai. The function has roots at every periodic point other then ai, and at that point the function's value is 1:

\[
l_i(z) := \prod\limits_{j=1 , j \neq i}^{n+m} \frac{(z-a_j)}{(a_i-a_j)}
 \]

(I hope I wrote the correct subscripts here in the text. I'll re-check a couple of times in the next days).


Offline Adam Majewski

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Offline Adam Majewski

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Re: Constructing polynomials with attracting cycles
« Reply #4 on: November 24, 2020, 05:55:48 AM »
https://arxiv.org/abs/2011.10935
On the formulas of meromorphic functions with periodic Herman rings
Fei Yang

Offline pauldelbrot

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Re: Constructing polynomials with attracting cycles
« Reply #5 on: November 24, 2020, 12:27:44 PM »
CooooOOOoooOooool! Thank you.

Image 1: supernova parameter space region containing the parameters for the rational example with a cycle of 2 Herman rings. What the paper calls a and b are set to the specified numbers; the image shows what Fei Yang calls the u-plane. There is a Mandelbrot set figure here, but the bulbs directly tangent to the cardioid all bear lesions on their borders, with a region of Herman mirror typically present. The Fei Yang example's u coordinate would be on the rim of the largest such bulb, in the lesion region about midway between its front bulb and the largest bulb left from there.

Image 2: the Julia set for a parameter on the edge of the period-4 bulb near the very bottom of image 1. There is a 4-cycle of Herman rings with a rotation number fairly close to 1/2.

I didn't even know, until today, that supernova was capable of producing Herman rings with period > 1. It seems it can in fact produce them at any period. Nifty!

UF parameters included, and formulas included in the parameter file. I used the hypernova formula, but with d = 0 it duplicates supernova while giving you all one extra knob to twiddle.