Some formula suggestion

[Alef]

Probably none is interested;) There are lots of formulas alredy.

But

I hade no idea how to call this but in age old Fractal Explorer it was under my name. Then I decided Deco.
In the zooms it looks like mandelbrot set but in each level of zoom stalks are surounded by miriads of sattelites which looks like julia sets and zooming in each sutch satelite reveals more satelites and minibrot at the center. For Fractal Explorer this was writen in pascal using two real numbers x, y but it was so long time agou that the code is lost. Alsou sinh must be complex function.

MalinovskyDecoFractal {
; z=z*sinh(z)-c*c or  z=z^(n-1)*sinh(z)-c^n
; Had written this formula more than 5 years agou for Fractal Explorer.
; with my name but deco sounds nice.
; Mandelbrot like set with sinh function.
; Features small julia satelites flying around the main set and
; more julias around julia each with mandelbrot at it's core + leave pattern.

(For me) it would be interesting to explore this with slopes. If it is initialised not with z= 0 but with z= pixel (c) in the corners instead of small mandelbrots it will have lots of weird fractals.
<a href="https://www.youtube.com/v/CDlJjy9TybA&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/CDlJjy9TybA&rel=1&fs=1&hd=1</a>

This formula were found and explored here in the forums by "Fracmonk". It have different power shapes on same set.

Multipowerbrot:
; Formulas by Fracmonk aka Jeffrey Barthelmes
; from this thread
; http://www.fractalforums.com/new-theories-and-research/is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/
; Z=(((((Z^2xC) +1 )^2) -1 )^2) -1 is posted:
; http://www.fractalforums.com/index.php?topic=4881.msg25998#msg25998
; Z=(((((Z^3)xC+i)^3)+i)^3)+i is posted:
; http://www.fractalforums.com/index.php?topic=4881.msg27524#msg27524

In that thread there are lots Fractint zoom pictures of this "Is there anything novel left to do in M-like escape-time fractals in 2d?" was before a perturbation theory.






This one is something like mandelbrot foam:

Devaney {
; z=  z^n + C / (z - a)^d
; whose versions was much researched
; by Robert L. Devaney
; julia sets = Sierpinski carpet.
; http://math.bu.edu/people/bob/papers.html
; 09.09.2014.
; 20.09.2014. finaly normal critical points for mset.
; thanks to http://www.fractalforums.com/index.php?topic=18526.30
; not much compatibility with previous critical points, sorry.

Initial conditions
 z=(@powerd*c/@powern)^recip(@powern+@powerd)
Iteration
z= z^@powern + c / (z - @talis)^@powerd

Basic form of pow2
Initialisation
 z=(2*c/2)^(1/(2+2)
Iteration
z= z^2 + c / (z )^2

Of these formulas UF users seems to liked the most z=z*sinh(z)-c^2 and Devaney formula, but I think devaney is not very zoomable. Throught around mandelbrots in these nets are something like julia sets.
I don't think they will be "perturbed".

[claude]

I added some numbers to this edited quote:

(1) ; z=z*sinh(z)-c*c or  z=z^(n-1)*sinh(z)-c^n
(2) ; Z=(((((Z^2xC) +1 )^2) -1 )^2) -1 is posted:
(3) ; Z=(((((Z^3)xC+i)^3)+i)^3)+i is posted:
(4) ; z=  z^n + C / (z - a)^d

(1) is problematic, because of the sinh() - the mpc library does have arbitrary precision complex sinh, but I don't think anyone has found perturbation formulas for the transcendental functions, and Kalles Fraktaler doesn't render without perturbation. If you only want slopes and can live with shallow zooms, maybe try Fragmentarium - it has complex sinh in recent versions, and shaders can use dFdx() and dFdy() to approximate derivatives for slopes.

(2) and (3) are problematic because of the multiplication by C, see http://www.fractalforums.com/kalles-fraktaler/creating-custom-fractals-on-kalles-fraktaler/  Only adding C works.

(4) is problematic similar to (1) because of the division (no perturbation formula to my knowledge), and probably for the same reason as (2),(3) too

Sorry!

[Alef]

Yes, I wanted slopes. Fragmentarium have another problem, it don't works on my pc
Then nothing. Thanks for explanation.
I realy hadn't looked deeper on what is perturbations.

[Kalles Fraktaler]

Yep, c must only be added so I don't think any of your formulas work with perturbation.