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### Author Topic:  feigenbaum stretch  (Read 709 times)

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#### claude

• Uploader
• Posts: 1937
##### feigenbaum stretch
« on: November 05, 2018, 09:51:59 PM »
feigenbaum stretch

https://fractalforums.org/index.php?action=gallery;sa=view;id=1653

Mandelbrot set with Feigenbaum point stretched off to -infinity at the left by exponential / logarithmic transformation

Linkback: https://fractalforums.org/index.php?topic=2112.0

#### Adam Majewski

• Fractal Frankfurter
• Posts: 531
##### Re: feigenbaum stretch
« Reply #1 on: November 05, 2018, 11:07:27 PM »
Cool
Can you add more components to the image ? (until image get dark)
Where is the code ?
Can you describe algorithm ( transformation) ?

#### claude

• Uploader
• Posts: 1937
##### Re: feigenbaum stretch
« Reply #2 on: November 06, 2018, 12:09:51 AM »

#### gerrit

• 3f
• Posts: 2448
##### Re: feigenbaum stretch
« Reply #3 on: November 06, 2018, 04:26:29 AM »
Transform is $$c \leftarrow e^c + c_0$$ with $$c_0 = −1.401155189092051.$$
How do you find $$c_0$$, or other Feigenbaum points?
It's a fun map. Here's one for the Fbm point defined by going to period 3 mini, then it's relative period 3 mini, etc.

#### claude

• Uploader
• Posts: 1937
##### Re: feigenbaum stretch
« Reply #4 on: November 06, 2018, 05:39:58 AM »
How do you find $$c_0$$, or other Feigenbaum points?

https://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/bin/m-feigenbaum.c estimate location of period 2P atom using size of period P island, use Newton's method to find its nucleus and size more precisely, repeat
https://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/bin/m-feigenbaum3.c estimate location of tip of antenna (renormalized from -2+0i) of period P island using size estimate, from there find nucleus of period 3P island using Newton's method, repeat

#### gerrit

• 3f
• Posts: 2448
##### Re: feigenbaum stretch
« Reply #5 on: November 07, 2018, 04:30:10 AM »
https://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/bin/m-feigenbaum.c estimate location of period 2P atom using size of period P island, use Newton's method to find its nucleus and size more precisely, repeat
https://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/bin/m-feigenbaum3.c estimate location of tip of antenna (renormalized from -2+0i) of period P island using size estimate, from there find nucleus of period 3P island using Newton's method, repeat
Neat, and it also shows why there are uncountably many Feigenbaum points.

#### Adam Majewski

• Fractal Frankfurter
• Posts: 531
##### Re: feigenbaum stretch
« Reply #6 on: October 17, 2019, 07:12:19 PM »

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