All periodic bulbs as point attractors.....

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Offline FractalDave

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« on: December 21, 2018, 01:13:50 PM »
Hi all, apologies to the math geeks if this is already common knowledge....I just found (when playing with the math behind Laguerre's method with a view to implementing similarly to Newton's) that the z^p+c derivative Set render is identical to the plain z^p+c but ALL the periodics are converted to point attractors.
In the image below the convergent areas are bailed out using the standard z-zold magnitude method and coloured using plain smooth iteration (as is the divergent area), the main original point attractor cardioid is coloured perfectly but the others have some issues (though they look OK till you examine in more detail, specifically the smooth fractions are not 0 to 1 on each iteration band and position/scale appears to vary with iteration depth, I'm looking into it).
See images, you can see the banding on the zoom:
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Offline FractalDave

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« Reply #2 on: December 21, 2018, 03:15:18 PM »
the derivative of the limit cycle is <1 in the interior.  see https://mathr.co.uk/blog/2013-04-01_interior_coordinates_in_the_mandelbrot_set.html https://mathr.co.uk/blog/2014-11-02_practical_interior_distance_rendering.html

Thanks that's damn useful and interesting !!

Also......DUH!!....I really should think about the maths as carefully as the programming, every single iterated derivative of In(z^p+c) will have zero as a root because in the derivative all the plain +c^q vanish !!
Of course that doesn't necessarily mean all the convergent attractors of the derivative are zero, but they do seem to be - I altered the colouring to use the fixed single point attractor bailout test using zero and bingo - it still works ;) Unfortunately still issues with the smooth iteration fraction on all but the main cardioid......however if that can be fixed it should be faster than having to use the periods/roots at least for getting a DE (certainly a deltaDE, possibly an analytical one) and maybe even a faster but equally good image mapping method ;)


Edit: Actually if zero is a root of every iteration then maybe zero is always the convergent attractor ? Anybody ?
« Last Edit: December 21, 2018, 03:39:21 PM by FractalDave »

Offline quaz0r

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« Reply #3 on: December 21, 2018, 09:45:09 PM »
im curious to follow any mandelbrot discussions but im not a math scientist so lots of times im not sure what you guys are talking about from the language you use   :(  what do you mean by "derivative Set render" ?  are you doing escape-time coloring on the derivative?

Quote
the convergent areas are bailed out using the standard z-zold magnitude method
not sure what this means either.  you check |zn|-|zn-1| ?  checked against what condition?

i was curious to find a simpler interior coloring method myself, as the distance algorithm with finding roots and periods seems complicated at best, perhaps unreliable at worst?  so i found this

https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set_interior#The_Lyapunov_Exponent

the simplicity of summing ln|2z| made me hopeful at first, though practically it is quite slow, probably due to taking the log every iteration, and also i found that whatever maxIter you need to make the exterior look good is also what you need to make the interior look good with this method.  it also suffers from the same problem as escape-time coloring for the exterior, with the interior of each bulb and each minibrot appearing denser and denser the deeper you go, needing to be normalized or something to look good.

Offline FractalDave

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« Reply #4 on: December 22, 2018, 03:33:12 AM »
im curious to follow any mandelbrot discussions but im not a math scientist so lots of times im not sure what you guys are talking about from the language you use   :(  what do you mean by "derivative Set render" ?  are you doing escape-time coloring on the derivative?

Yes - the derivative basically as it's calculated for anlytical DE using the implementation of the chain rule for the derivative of f(g(x)) where f(x) is say g(x)^p+c and the value of g(x) is current z.
So f'(g(x)) is p*z^(p-1) and g'(x) is the derivative from the previous iteration, so on each iteration the new derivative is: dz = dz*p*z^(p-1) AND new z = z^p+c as normal
What I'm doing is rendering the escape-time fractal for dz instead of z and it turns out that doing this converts all convergent areas to point attractors at least for z^p+c, and in fact more than that they beome zero attractors.

"not sure what this means either.  you check |zn|-|zn-1| ?  checked against what condition?"

Nearly - magnitude of z-zold (or dz-dzold) i.e. |z-zold| against a "small bail" parameter i.e. if magnitude of z-zold less than smallbail then bailout - the general method for point attractor bailout when the fixed point attractor value is unknown.

Offline claude

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« Reply #5 on: May 15, 2019, 12:33:48 PM »
I think you have rediscovered https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/Mandelbrot_set#The_idea

which also works for Burning Ship when you use the Jacobian derivative matrix w.r.t. (x_1, y_1).

Offline hgjf2

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« Reply #6 on: May 17, 2019, 09:23:25 PM »
wow! The nuances of green in red at this burningship look like Lyapunov fractal
 :thumbs_up_by_craig_m: :thumbs_up:


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