Comparsion of the functions of the Mandelbrot set and a bifurcation diagram.

The function for the Mandelbrot-set is:

Z_n+1 = Z_n² + C

The function for the (standard) bifucation-diagram is:

X_n+1 = A · X_n · (1 - X_n)

The Mandel-formula is using a complex-number format, the bifurcation is using real numbers. But the formulas can be altered to the diffrent form and used to write the Mandel as a bifurcation and the bifurcation as a complex escape-time fractal.

The real number Mandel formula is:

X_n+1 = X_n² + A

And the complex bifurcation function is:

Z_n+1 = C · Z_n · (1 - Z_n)

The image above shows all four variations, to the left the Mandelbort formula as a bifurcation diagram in white and also the complex "escape-time" visulation of the complex formula in blue, to the right, the same for the (standar, there are a lot of those =) bifurcation formula and as we can see, these formulas are strongly related.

Image: User:Solkoll. Public domainPublic domainfalsefalse This work has been released into the public domain by its author, Solkoll. This applies worldwide. In some countries this may not be legally possible; if so: Solkoll grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. More fractals from my tools: Image:Fractal dimension.jpg Image:Harter-Heighways dragon curve (IFS).jpg Image:Koch curve (L-system construction).jpg Image:Koch curve (IFS).jpg Image:Koch snowflake (RGB-CMY).jpg Image:Lévy's C-curve (IFS).jpg Image:Lévy's dragon curve (IFS).jpg Image:Mandelbrot inside (64 iters B&W).gif Image:Mandelbrot inside (64 iters colour).gif Image:Menger sponge (2D).jpg Image:Julia set (C = 0.285, 0.01).jpg Image:Julia set (C = 0.285, 0.01) II.jpg Image:Julia set (Detail 01).jpg Image:Julia set (Detail 02).jpg Image:Julia set (highres 01).jpg Image:Julia set (highres 02).jpg Image:Julia set (Rev formula 01).jpg Image:Julia set (Rev formula 02).jpg Image:Julia set (Rev formula 03).jpg Image:Julia set (Rev formula 04).gif Image:Reversed Julia set C = ( 0.4 0.3 ).gif Image:Sierpinski square.jpg Image:Sierpinski triangle (blue).jpg Image:Sierpinski triangle (RGB).jpg Image:Solkoch.jpg All fractals are from self-written tools - linear IFS images from my: "2D IFS Studio", non-linear (rev Julia) from "3D RJIFS", (that also writes 2D rev Julias), escape time Z² fractals from "Julia dream", the rest are from small "hacks". See also: Solkoll & Solkoll 3D This image was uploaded in the JPEG format even though it consists of non-photographic data. This information could be stored more efficiently or accurately in the PNG or SVG format. If possible, please upload a PNG or SVG version of this image without compression artifacts, derived from a non-JPEG source (or with existing artifacts removed). After doing so, please tag the JPEG version with {{Superseded|NewImage.ext}} and remove this tag. This tag should not be applied to photographs or scans. If this image is a diagram or other image suitable for vectorisation, please tag this image with {{Convert to SVG}} instead of {{BadJPEG}}. If not suitable for vectorisation, use {{Convert to PNG}}. For more information, see {{BadJPEG}}.