Some Mandelbrot foam math.
Iteration is (with q a complex constant, c the pixel parameter)
\( w \leftarrow qw/z,\\z \leftarrow z^2+w^2 +c. \)
To determine critical point (to start from) both eigenvalues of the Jacobian
\( J=\begin{pmatrix} q/z & -qw/z^2\\2w & 2z\end{pmatrix} \)
should be 0.
\( 0 = det(J) = (z^2+w^2)2q/z^2 \) gives \( w = i z \) which makes at least one eigenvalue 0.
Substitute this in the formulas for the eigenvalues \( \lambda \) (using a symbolic calculator)
\( \lambda = (q + 2z^2 \pm (q^2 - 8qw^2 - 4qz^2 + 4z^4)^{1/2} )/(2z) \)
gives
\(
\lambda =(q+2z^2 \pm  (q+2z^2))/(2z) =0
 \)
so \( z = \sqrt{-q/2}. \)

Next I try to find if a stable fixed point (period 1 "mini") exists. Conditions are
\( w = qw/z,\\z=z^2+w^2+c,\\\max{|\lambda|}\leq 1. \)
We get
\( z=q\\w=\sqrt{q-q^2-c}. \)
Substitute this in the formula for the eigenvalues gives
\( \lambda = a \pm b \) with \( a=q+1/2 \) and \( b(c)=\sqrt{(4*q^3 + 4*q^2 - 7*q + 8*c)/q}/2. \)
Imposing \( |\lambda|\leq 1 \) on the "plus" formula gives
\( a+b = r e^{i\theta} \) with \( 0\leq r\leq 1 \) and \( \theta \) an angle.
The corresponding \( c \) can be found by solving \( b(c) =r e^{i\theta}-a  \) for \( c \).
The condition on the other eigenvalue is now
\( |a-b|=|2a-r e^{i\theta}|\leq1 \) which can be satisfied only if \( |a|\leq 1 \).
So there exists a period 1 region if \( |2q+1|\leq 1 \).
Maybe a good parametrization is \( q=(1+\rho e^{i\phi})/2 \) which should give period 1 regions if \( \rho<1 \).

Not sure if I made a mistake, for there seems to be nothing special happening at \( \rho=1 \) when I make images.
Also I'm not sure how to now get the theoretical shape of the period 1 region. in general.

Special case \( q=-1/2 \) (so a=0) is easier. Solving \( b(c)=r e^{i\theta} \) gives
\( c = r e^{i\theta}/4-1/2 \), which is a circle of radius 1/4 around -1/2.

Something's not quite right though, because the picture below shows almost that circle, but little pieces are chipped out at the edges. It's not an insufficient iteration issue. Anyone spot a mistake?
 

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/mandelbrot-foam-math/1665/

one set of fixed points is: w = 0, z = z^2 +c, so c is the m-set-alike cardioid
for w != 0, and q = -1/2, I get your result (up to sign) of c = -1/2 - t^2/4
I was assisted by maxima

re the chips, what is the other eigenvalue like at those points? the other eigenvalue should be 0 afaict.  not sure what is happening as I haven't got software to check visually..., I can reproduce this behaviour in Fragmentarium
guess: maybe the circle is too big, and the chips are at the real 1/4 radius? this could be caused by over-eager periodicity detection?
guess2: you are of course doing the iterations in parallel, not updating w before updating z with the new w?

yes I reproduced the chips.  sorry for doubting

yes I reproduced the chips.  sorry for doubting

We seem to be crossposting, though that used to mean something else in the past.
Doubting is good and has no downsides.

The choice of \( z_0=\sqrt{-q/2} \) is not important, any value with a w that makes the determinant 0 (\( w=iz \)) gives the same image. This makes only the second eigenvalue 0.

If you make only the first eigenvalue 0, take any nonzero \( z_0 \) and set
\( w_0 = \sqrt{  (q^2-4qz_0^2+4z_0^4-(q+z_0^2)^2)/(8q) }  \), the result is different and also nice, but it now does depend on \( z_0 \), so there another parameter to play with.

Below q=-1/2 with both choices, \( z_0=1 \) for the second method.

Added: I must be doing something wrong with this other "choice"; if an eigenvalue is 0 the determinant must be zero but then we must have \( w=iz \), so the second eigenvalue must be 0. Pictures look nice anyways...

Distance estimator.
I've been using the following method for DE which seems to produce good results.
\(
dw = q(dw/z - w/z^2 dz)\\dz = 2w*dw +2zdz+1\\ D = |z|log|z|/|dz|
 \)
Ghost dots appear which seem to be local maxima of the DE, so sometimes you try to zoom into one of those before you realize there is nothing there.

it seems the w iteration also needs high precision and it can't be molded for PT as far as I can see as it's non-polynomial.

This is what I get:

W := q W / Z
W + w := q (W + w) / (Z + z)
w := q (Z w - z W) / (Z (Z + z))

Z := Z^2 + W^2 + C
Z + z := (Z + z)^2 + (W + w)^2 + (C + c)
z := 2 Z z + z^2 + 2 W w + w^2 + c

I haven't tried implementing it, so maybe there is a fatal flaw.  See also: https://mathr.co.uk/blog/2018-03-12_perturbation_algebra.html

reminds me of this random image