 • January 23, 2021, 08:37:10 PM

### Author Topic:  Mandelbrot foam math  (Read 1562 times)

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#### gerrit ##### Mandelbrot foam math
« on: July 29, 2018, 10:03:59 PM »
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?

« Last Edit: July 30, 2018, 11:33:39 PM by gerrit, Reason: Corrected errors/added shape analysis »

#### gerrit ##### Re: Mandelbrot foam math
« Reply #1 on: July 30, 2018, 11:34:37 PM »
Corrected errors...

#### claude ##### Re: Mandelbrot foam math
« Reply #2 on: July 31, 2018, 03:57:26 AM »
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?

« Last Edit: July 31, 2018, 04:37:00 AM by claude, Reason: reproduced »

#### gerrit ##### Re: Mandelbrot foam math
« Reply #3 on: July 31, 2018, 04:15:42 AM »
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...
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?
The eigenvalues for $$q=-1/2$$ are $$\pm \sqrt{-4c-2}$$.

Thanks for the observation that the M-set cardioid is a stable fixed point. The fact that it is never seen must be due to the initial conditions; this orbit is not reachable from the critical point. So maybe that explains the chips: those areas do have a stable fixed point but it is not reachable from the initial condition.

Edit in response to your edit: No, I zoomed into the boundary of the "circle" and verified the chips are actually that: regions where there is a stable fixed point, but it is not reachable. Calculation is just calculating every pixel, and yes, I took care of using correct w in the z update.

#### claude ##### Re: Mandelbrot foam math
« Reply #4 on: July 31, 2018, 04:37:55 AM »
yes I reproduced the chips.  sorry for doubting

#### gerrit ##### Re: Mandelbrot foam math
« Reply #5 on: July 31, 2018, 04:42:21 AM »
Here's an image for q=-1/2 centered at -0.499963345735437+0.249685522638552 * i, i.e., in the "circle".

BTW PT should be easy for this beautiful fractal by pauldelbrot. w can remain normal precision, just perturb z. I tried to implement it in UltraFractal but it does not work; I think it only supports PT in one variable.

#### gerrit ##### Re: Mandelbrot foam math
« Reply #6 on: July 31, 2018, 04:44:35 AM »
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.

#### gerrit ##### Re: Mandelbrot foam math
« Reply #7 on: August 01, 2018, 04:29:48 AM »
Thanks for the observation that the M-set cardioid is a stable fixed point. The fact that it is never seen must be due to the initial conditions; this orbit is not reachable from the critical point. So maybe that explains the chips: those areas do have a stable fixed point but it is not reachable from the initial condition.
I verified (by computing single orbits) that a point near the edge of the circle that escapes, converges when (w0,z0) is sufficiently close to where the attractor is.

#### gerrit ##### Re: Mandelbrot foam math
« Reply #8 on: August 01, 2018, 05:10:19 AM »
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...
« Last Edit: August 01, 2018, 06:10:39 AM by gerrit »

#### gerrit ##### Re: Mandelbrot foam math
« Reply #9 on: August 08, 2018, 05:16:55 AM »
Connection to the Mandelbrot set.
The M-foam iteration
$$w \leftarrow qw/z,\\z \leftarrow z^2+w^2 +c.$$
can be rewritten in a suggestive form by setting $$z = u+v$$, which gives the equivalent iterations
$$u \leftarrow u^2 +c,\\w \leftarrow qw/(u+v),\\v \leftarrow v^2+2uv+w^2.$$
So $$u$$ obeys the normal Mandelbrot iteration, and the other variables do not depend explicity on $$c.$$
The escape condition now is $$|z \equiv u+v|<R$$ with $$R$$ the (large) escape radius.

This suggest a fast way for deep zooming: just use Mandelbrot perturbation for the u-orbit and calculate v and w in normal precision (as it does not involve c). Problems arise when u escapes but $$z \equiv u+v$$ does not. There are however not many point where this happens and they are perhaps "uninteresting" (all disconnected Juliaset-like stuff) anyways. Interesting points seem to be in the interior of the associated M-set.

Attached an overlay of the M-set on the M-foam for $$q=1/2$$. Larger values of q only have disconnected stuff, and smaller values have the interesting stuff happening inside the associated Mandelbrot shape. Second image shows a detail with a "tongue" that is not in the M-set but in the M-foam set. This is where both u and v get very large but cancel.

#### gerrit ##### Re: Mandelbrot foam math
« Reply #10 on: August 08, 2018, 05:22:33 AM »
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.

#### gerrit ##### Re: Mandelbrot foam math
« Reply #11 on: August 14, 2018, 04:46:58 AM »
It seems PT does not work using the ideas presented previously. I tried perturbing only z in various ways, but 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.

I made some obvious generalization of the original which produces nice images. Basically use different powers ($$p_1, p_2, s_1, s_2$$), normally $$(2,2,1,1)$$, and compute critical point.
General form is
$$z_0 = \sqrt{-q/2}\\ w_0 = (- z_0^{p_1} p_1 s_1/(p_2 s_2) )^{1/p_2}\\ dw = 0\\ dz = 0\\ dw \leftarrow s_1 w^{s_1-1}/z^{s_2} dw -s_2 w^{s_1}/z^{s_2+1} dz\\ dz \leftarrow p_1 z^{p_1-1} + p_2 w^{p_2-1} + 1\\ w \leftarrow q w^{s_1}/z^{s_2}\\ z \leftarrow z^{p_1} + w^{p_2} + c$$
The choice of $$z_0$$ (which makes all eigenvalues zero) does not seem to matter, as long as $$w_0$$ relates in the indicated way.

A nice variant is to use a non-critical initial point
$$z_0=c\\w_0 = r c$$
with $$r$$ a small number.
This produces Mandelbrot-like shapes with the interior filled in. Minibrots are also filled in, but $$r$$ needs to be very small before they disintegrate.

You can also add $$c$$ to the $$w$$ update ($$dw$$ formula needs an extra +1 then) without affecting the critical point. This variant does not seem too interesting, except when combined with the alternative starting point and adding $$rc$$ to the $$w$$ update (and of course $$+r$$ to the $$dw$$ update).

#### claude ##### Re: Mandelbrot foam math
« Reply #12 on: August 14, 2018, 05:30:51 AM »
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

#### gerrit ##### Re: Mandelbrot foam math
« Reply #13 on: August 14, 2018, 08:00:15 AM »
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
I wrote down those same formulas, but gave up realizing those reference orbits are almost always going to escape early giving many glitches so to try it some glitch detection/correction is needed. Just laziness, maybe it will work.

#### v

• Fractal Phenom
•    • Posts: 54 ##### Re: Mandelbrot foam math
« Reply #14 on: August 23, 2018, 09:28:28 AM »
reminds me of this random image

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