Image:Mandel-bifurk.jpg

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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 different 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.

This image has been (or is hereby) released into the public domain by its creator, Solkoll. This applies worldwide. In case this is not legally possible, the creator grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. Subject to disclaimers.

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.

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Image: User:Solkoll. This image 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. العربية | Български | Deutsch | Ελληνικά | English | Español | Français | Italiano | 日本語 | 한국어 | Polski | Русский | Српски / Srpski | Svenska | தமிழ் | Українська | ‪中文(繁體)‬ | ‪中文(简体)‬  +/- 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

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