Talk:Mandelbrot set/Archive 4

Ellipse's

Every time I post this kind of stuff somebody comes in and deletes it. So, I'll leave this here for you folks and let you decide.

Ellipse Positive Set, x^2+x*y+y^2=1, <(1/3)*sqrt(3)*cos(ϑ)-sin(ϑ), (1/3)*sqrt(3)*cos(ϑ)+sin(ϑ)>: z^1 = <x+x_0, y+y_0> z^2 = <x^2-y^2+x_0, y*(y+2*x)+y_0> z^3 = <x^3-3*x*y^2-y^3+x_0, 3*x*y*(x+y)+y_0> z^4 = <x^4-6*x^2*y^2-4*x*y^3+x_0, 4*x^3*y+6*x^2*y^2-y^4+y_0>

Ellipse Negative Set, x^2-x*y+y^2=1, <cos(t)-(1/3)*sqrt(3)*sin(ϑ), cos(ϑ)+(1/3)*sqrt(3)*sin(ϑ)>: z^1 = <x+x_0, y+y_0> z^2 = <x^2-y^2+x_0, y*(-y+2*x)+y_0> z^3 = <x^3-3*x*y^2+y^3+x_0, 3*x*y*(x-y)+y_0> z^4 = <x^4-6*x^2*y^2+4*x*y^3+x_0, 4*x^3*y-6*x^2*y^2+y^4+y_0> Example of z^2 from negative set. https://www.khanacademy.org/computer-programming/z2-ellipse-normalized/6285584115646464 Left click zooms in, right click zooms out.

Mixed Circle and Ellipse sets mixed. <x^2-y^2+x0^3-3*x0*y0^2+y0^3, 2*x*y+y_0+3*x0*y0*(x0-y0)> Here is an example of a mixed set: https://www.khanacademy.org/computer-programming/cn2e03-magnitude/5661509447008256 Changed the distance function to an area function, x0*y1+x1*y0. This is the area between the vector <x0, y0> and the vector <x1, y1>. If I need to explain this further, lemme know.

This one, it was an accident. But give it a chance. Zoom in a bit. If you can find it off to the center left, there is a set of lungs beside the heart. https://www.khanacademy.org/computer-programming/cn2ie32-magnitude/5411191639457792

X-Rated <(x0+x1+y0+y1)*(x0+x1-y0-y1)+x0, (2*(y1+y0))*(x1+x0)+y0> https://www.khanacademy.org/computer-programming/x-rated-mandelbrot/5791655344685056 — Preceding unsigned comment added by Tejolson (talk • contribs) 13:16, 24 November 2020 (UTC)

(z^2)^2 Mandelbrot: <(x0^3-3*x0*y0^2+x1^3-3*x1*y1^2)^2-(3*x0^2*y0+3*x1^2*y1-y0^3-y1^3)^2, (2*(3*x0^2*y0+3*x1^2*y1-y0^3-y1^3))*(x0^3-3*x0*y0^2+x1^3-3*x1*y1^2)> https://www.khanacademy.org/computer-programming/cn3c32-magnitude-color-mix-variation/5412316920889344 Oh, that's not a (z^2)^2 Mandelbrot? I must have forgotten something... <(x0^3-3*x0*y0^2+x1^3-3*x1*y1^2)^2-(3*x0^2*y0+3*x1^2*y1-y0^3-y1^3)^2, (2*(3*x0^2*y0+3*x1^2*y1-y0^3-y1^3))*(x0^3-3*x0*y0^2+x1^3-3*x1*y1^2)> https://www.khanacademy.org/computer-programming/c_n3ic32c-magnitude-color-mix-variation/5092117646622720 Crap, it's a star for some reason. Almost a star fish quality to it. It's suppose to be the original z^2 Mandelbrot Set. Oh, and I used a mixed color variation. It's bed time. I gotta go hunting in 3 hours. — Preceding unsigned comment added by Tejolson (talk • contribs) 08:03, 24 November 2020 (UTC)

Proposal for subsection of Generalization

Dear community. I fully understand that Wikipedia is not the place to promote one’s own work. I therefore open this discussion here instead of editing the page. I recently published a peer-reviewed scientific article in the journal Chaos. (Chimeras confined by fractal boundaries in the complex plane, Chaos 31, 053104 (2021)). In this article I study the fractal sets generated by a network of four coupled quadratic maps. Below I summarize the main findings and provide references. In my view, this could be considered as a generalization of the Mandelbrot set to four coupled maps. It might therefore fit into the section 'Generalization'. Looking forward to hearing your opinions.

In a nutshell: In analogy to the dichotomous behavior of individual quadratic maps, the states of networks of coupled identical quadratic maps either remain bounded or diverge to infinity. Beyond that, the bounded states can further be sub-classified with regard to their synchronization. In my article, I study the case of a minimal two-population network of two pairs of each two quadratic maps. In this case, one can find full desynchronization of all four maps, full synchronization of all four maps, or different states of partial synchronization. Among the states of partial synchronization are so-called chimera states, for which one pair of maps synchronizes and the maps of the other pair remains desynchronized. In these images (iterative zoom, details of one zoom step), gray colors are used for complex c at which the network states diverge, where lighter gray indicates slower divergence. Different non-gray colors indicate the different states of synchronization. The resulting fractal patterns seem in general less filigree, more disordered but also fuller of variety than the Mandelbrot set.

References: The printed article at the journal page can be found here (open access only until 17th of May). The reprint of the article at the homepage of my university can be found here (identical content and open access). The source code used to run my simulations is provided here.

--Cyclingralph (talk) 20:56, 6 May 2021 (UTC)