Julia and parameter-space images of polynomials

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Offline gerrit

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« Reply #45 on: July 27, 2019, 07:01:41 AM »
Selecting the "not-c" parameter(s) to be close to the a "critical value" where there ceases to be a region with nothing escaping seems a good heuristic to get interesting images.

Below again for \( f = z^3-3a^2z+c \) with \( a=0.59 e^{1.2 \pi i} \), with exterior colored by distance estimation.
Blue in first image is where both orbits stay bounded. Second image zoom into center region, white is all orbits bounded.

Offline gerrit

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« Reply #46 on: July 27, 2019, 10:46:48 PM »
 Some analysis, I think the "marcm200 set" contains |a|^2<1/3.
\( f(z) = z^3-3a^2z+c \), critical orbits \( z_0 = \pm a \).

Main bulb (period 1) must have
\(
z = z^3-3a^2z+c\\
|f'(z)=3(z^2-a^2)| < 1
 \)
which gives
\(
z^2 = a^2+\frac{1}{3}r e^{i \theta}\ \ (*)\\
c=(3a^2+1)z-z^3
 \)
with 0<r<1 and angle going around 360 degrees.

Let's look at the nucleus of the \( z_0=a \) orbit. We have \( c = (1+2a^2)a \) and for that c we have indeed
\( f(a) = a \), the nucleus, with zero derivative. This nucleus will have the other critical orbit also bounded if \( z_0=-a \)
lies in the domain of attraction. Formula (*) represents a circle of radius 1/3 centered on a^2, and if it does not include origin, sqrt will have two disconnected parts so the +a nucleus can't be reached from -a. For r<1/3 sqrt leads to 1 region and both orbits attract. So  |a|^2<1/3 is in the set.

Below a detail for \( a=\sqrt{1/3}e^{0.0946 \pi i} \) using only 1 critical orbit, and a Julia set of the region.

Offline marcm200

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« Reply #47 on: July 29, 2019, 07:22:15 PM »
Some analysis, I think the "marcm200 set" contains |a|^2<1/3.

That's great.

I wasn't quite that successful and could only computationally verify by Figueiredo's trustworthy Julia set algorithm that all \( a \in [0..0.25] \times [0i..0.25i] \) are in the set.

I computed a trustworthy Julia set with interval arithmetics for \( a \) in the above region using your formula \( f(z) = z^3-3a^2z+c \) for \( c=(6308234 + 0i) * 2^-25 \approx 0.188 +0i \) (see trustworthily computed image 1 below). The red and blue squares are the regions of the critical points \( a \) and \( -a \) and lie completely immersed in the bounded, black-colored region (white is, as always with the TSA, definite escaping starting points and gray is undecided at the current refinemnt level 12).

All orbits starting in those two boxes remain bounded for every choice of \( a \) in the given intervall, so none of the two critical point's orbit is ever escaping.

The seed was found through computing by point-sampling several c-parameter space sets for some choice of \( a \) as the corners of small squares where the as-well-point-sampled brute-force image of the "simpler set" showed interior.

Afterwards I took the intersection of the c-parameter spaces for \( a=0+0i, 0+0.25i, 0.25+0i, 0.25+0.25i \), hoping something would be left. And at that width 1/4 there was (see point-sampled image 2 below). I chose a seed value from deep
within that intersection and again hoped that this would still work in the interval arithmetics trustworthy case, which it did.

(I'm quite sure c=0+0i would've also worked but I took the value the code suggested me by sliding a 32x32 square over the black region and taking its middle point which resulted in the above c-value).


Offline gerrit

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« Reply #48 on: August 01, 2019, 04:27:53 AM »
Modest zoom (1e5) into interior of main "blob" of 78th degree polynomial. Blob is where not all 77 critical orbits diverge.
Coloring is 1-77: the number of escaping orbits at the pixel.

Offline marcm200

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« Reply #49 on: August 16, 2019, 08:20:51 AM »
A c-parameter space M-set for z5+A*z+c with A being (-34209751+0i)*2-25.

There's quite a lot going on here: distorted mini M-sets, some quadratic Julia set shapes,
objects greeting each other and a large pink tentacle that is held by a clip (upper left).

Colored by fate of critical orbit (16 combinations in theory, black means all bounded),
critical points found numerically by Sutherland et. al's universal set of Newton starting points.

Offline gerrit

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« Reply #50 on: August 16, 2019, 05:40:51 PM »
critical points found numerically by Sutherland et. al's universal set of Newton starting points.
Nice. Why? To test the Sutherland algorithm?

Offline marcm200

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« Reply #51 on: August 16, 2019, 06:02:48 PM »
Nice. Why? To test the Sutherland algorithm?
Yes, it's a test. I am starting to create (numerical) sets of universal points for some polynomial's degrees along the way (on a circle and on a square) and then reuse them afterwards if needed (final goal being to find parabolic periodic cycles in M-sets). Of course, in all practical sense, for small degrees like this image, it's not necessary at all and way overboard.

Offline marcm200

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« Reply #52 on: August 19, 2019, 08:01:45 PM »
"The impossible set"

This is a degree 5 polynomial Julia set with 399 (!) basins of attraction - which clearly is nonsense. But my (self-coded) software to numerically detect attracting cycles spot that out. If I recall correctly, there can only ever be "degree minus 1" attracting cycles.

But despite the obvious coding error, I still like the picture - and since it is quite regular in pattern, that mistake is somehow related to the inner structure of the Julia set, perhaps just numerical instability.

Offline claude

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« Reply #53 on: August 19, 2019, 08:03:43 PM »
are you counting immediate basins or preperiodic basins also?

Offline marcm200

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« Reply #54 on: August 19, 2019, 08:51:51 PM »
I intended to count the cycles bounded orbits were in at the end of max iteration. But I wanted to be too smart and have a fast algorithm so I checked for periodicity during iteration, not knowing whether an orbit escapes or not. The idea being that only the attracted orbits will come closer than my epsilon (1E-6, not small enough?) to a periodic point.

When I moved that check outside the loop when I definitely know whether an orbit escapes or not and only check the non-escaping ones, I got 2 attracting cycles, which sounds fasible. I attached a small image - very colorless, I liked the 399 one better. So I guess the end point of many iterations is closer to an actual periodic point than its first probable occurance early in the orbit (which I'm sure everyone except me was aware of).

But counting preperiodic basins would be interesting - take a cycle and check whether there is a finite set of entry points just before being trapped in the cycle (is that what you meant by preperiodic basins?)


Offline marcm200

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« Reply #55 on: Yesterday at 08:37:40 PM »
A degree 6 polynomial c-parameter space with two features depicted.

p(z)=z6+A*z+c with A=(-1.1930708339729698686+0i) (that strange number was randomly generated).

First image shows the fate of the critical orbits as usual, black being all 5 bounded (only c=0 is black).

Then I was interested in determining how many of the critical orbits for the underlying Julia sets go into a periodic cycle - and there in particular in how many different cycles. I read something about attracting cycles always attract the critical points, but I am not sure whether that meant they all go into different cycles each, or whether that's only true for the quadratic case.

Here are the first experimental results (code not yet checked, 4000 max it for cycle detection): The colored rectangles at the bottom show (from left to right) the colors for 0-5 different critical orbit cycles. A gray interior means that specific number of cycles was not encountered in the current resolution for any c value tested.

Background: I am searching for A/c values to get Julia sets with the maximum number of attracting cycles (degree-1), complicated shape and hence basins of attraction to color with the trustworthy Julia set algorithm and check the difference in interior emergence.


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