• October 20, 2021, 01:38:53 AM

### Author Topic:  Partial-abs burning ship  (Read 130 times)

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#### marcm200

• 3e
• Posts: 1072
##### Partial-abs burning ship
« on: October 13, 2021, 09:48:13 AM »
A few experiments with a partial burning-ship-like formula:
$f_{n,m}(z) := \tilde{z}^n\cdot z^{(d-n)} + A\cdot \tilde{z}^m\cdot z^{(1-m)} + c \\ \tilde{z} := |Re(z)| + i\cdot |Im(z)| \\$
here for the values
$d=3 \\ 0 \leq n \leq 3 \\ 0 \leq m \leq 1 \\$I followed the critical points of the pure polynomial form (n=0,m=0) in all other (n,m) combinations, although I do not know whether these orbits bear a meaning in the (partial) abs() cases.

Image below: black, escaping; white, all critical orbits are bounded; gray, only one orbit is bounded (but could be either of the two). A values were randomly generated and all (n,m) combinations computed.

Parameter A=-0.6435546875-0.947998046875*i:

2nd image: (n=0,m=1)
The 2-orbit-bounded part looks rather ragged. Overall there are many aberrations often seen in the lyapunov images: a lot of seemingly randomly set points of different escape fate than the surroundings. Looking only at one cp bounded gives in part the fractal bulb structure back, remnants of the non-abs version (lower left corner).

And, of course, there is that large escaping hole within a deep white region with its neighbourhood looking like a strange attractor.

The 1st image shows the full abs version (3,1) with a nice collection of tricorns and minibrots (arrow at top part), which is quite common for non-zero A to a more or less distorted degree. Interesting to see a tricorn-like shape in an abs formula rather than conjugated z.

#### marcm200

• 3e
• Posts: 1072
##### Re: Partial-abs burning ship
« Reply #1 on: October 14, 2021, 04:02:41 PM »
Looking at the escape behaviour near the (both critical points escaping) hole revealed an interesting orange wing (image below, upper part). It seems to form a continuation of the 1-cp-bound/1-cp-escaping gray wing. And a lot more of those gray parts behave the same. I haven't seen such a continuation into the escaping before.

• points with both critical orbits escaping are colored according to the faster escape time
• One cp escaping is uniformly gray, none escaping is black here (as opposed to the previous post)
• image colored by an inverse heat-map: cold colors for longer escaping, proportional to log(escape iter)
• lower left is a small overview as in the previous post, where black means both cps escaping and white both bounded.

#### marcm200

• 3e
• Posts: 1072
##### Re: Partial-abs burning ship
« Reply #2 on: October 15, 2021, 09:06:37 AM »
Using the / EDIT: modified / conjugate of z  in place of the split abs for the rational family
$f_{n,m}(z) := \bar{z}^n\cdot z^{(d-n)} + \frac{A}{\overline{z}^m\cdot z^{(d-m)}} + c \\ \text{EDIT: here}~\overline{z} := |Re(z)|-i\cdot Im(z)$

and conducting some random parameter A-space walks gave an interesting case in the c-parameter plane for $$d=2, n=2, m=1, A=-0.01513671875$$ when looking  at how many critical points (without poles) were bounded (not necessarily periodic or in the same cycle).

The image has a rich substructure, showing large regular regions (green/yellow, upper left white arrow) . Some parts at the boundaries to the all-escaping black seem to be chaotic in the sense, that  neighbouring pixels have different number of bounded critical points (lower right white arrow), but this might change in zooms.

• colors: 0 to 4 critical points bounded for black, red, yellow, green, blue
• Image cropped from the 2-square
• max it 5 000, arbitrary escape radius of 1024
« Last Edit: October 17, 2021, 12:30:25 PM by marcm200, Reason: parameter correction »

#### marcm200

• 3e
• Posts: 1072
##### Re: Partial-abs burning ship
« Reply #3 on: October 17, 2021, 12:39:37 PM »
Parameter correction for last post: It uses a modified conjugate version $$\overline{z} := z' := |Re(z)|-i\cdot Im(z)$$

-----------------
Here's a funny little image (c-parameter plane) - a red minibrot that seems to be chased by a big green boomerang, that, could I write, would make a nice setting for a shortstory, something in the line of "The mini and the army of three". But that's about as far as I got
$f(z) := (z')^2 + \frac{-0.01513671875}{z'\cdot z} + c$
with black, escaping; red, one critical orbit bounded; green, 3 cps bounded.

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