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Author Topic:  Higher order M sets  (Read 125 times)

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Offline mcneils

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Higher order M sets
« on: February 22, 2021, 03:04:07 PM »
Not sure if this is an old topic, but I'll ask anyway. I've seen many examples of 'higher order' Mandelbrots (z^2 + c), where 'z' alone is increased to higher orders (e.g., z^4 + c). I have found that the antiderivatives make for some interesting regions, with many 'stable' areas not locally connected with the main set. The 8th antiderivative is shown here, with increasing magnification. Does this imply that the common M set is a projection of higher order sets?


Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/higher-order-m-sets/4048/

Offline gerrit

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Re: Higher order M sets
« Reply #1 on: February 22, 2021, 05:53:25 PM »
Which antiderivative of what?

Offline mcneils

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Re: Higher order M sets
« Reply #2 on: February 22, 2021, 06:45:26 PM »
It is the 8th indefinite integral of z^2 + c. The equation, in Matlab syntax, is:
    z= z.^8/20160 + z.^6/720 + z.^5/120 + z.^4/24 + z.^3/6 + z.^2/2 +c.*z + c;
I used 50 iterations, and also increased the upper bound (infinity) from 2 to 50.

Offline marcm200

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Re: Higher order M sets
« Reply #3 on: February 22, 2021, 07:37:16 PM »
It is the 8th indefinite integral of z^2 + c. The equation, in Matlab syntax, is:
    z= z.^8/20160 + z.^6/720 + z.^5/120 + z.^4/24 + z.^3/6 + z.^2/2 +c.*z + c;
I used 50 iterations, and also increased the upper bound (infinity) from 2 to 50.

A few things I don't understand:

Why is it called the 8th? You need 6 repeated integrations to get from z^2 to z^8.

The indefinite integral of z^2+c is 1/3*z^3+c*z + some constant d. Integrating this repeatedly until the degree is 8 should give rise to monomials with a variable in the coefficient (c,d and the subsequent ones), there should especially be a c*z^6*some number. The repeated derivative of your formula goes down to z^2+1 and not z^2+c, however.

Which critical point(s) are you analyzing to get your images? (I find these types of polynomials quite complicated as the cps are dependent on c).

Offline mcneils

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Re: Higher order M sets
« Reply #4 on: February 22, 2021, 08:16:14 PM »
@marcm200,
You are correct on all points.  :-[ 

Offline marcm200

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Re: Higher order M sets
« Reply #5 on: February 22, 2021, 08:31:44 PM »
@mcneils:
I explored higher degree polynomials once, but the dimensionality of the parameter space went up too fast - and I usually start with brute force walks. Making the coefficients correlated to one another by using your integration idea would have been an interesting means (but I haven't thought of that back then).

Have you explöred Julia sets with your polynomial as well and found appealing shapes?

Offline gerrit

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Re: Higher order M sets
« Reply #6 on: February 23, 2021, 06:10:03 AM »
I think if you include the arbitrary integration constants you're just investigating general polynomials.
First problem is to (numerically) find the critical points, then somehow find a way to use the n-1 critical orbits to make images in parameter space. Or just do Julia sets, but they are best found by picking in parameter space.

I think there is a thread on this stuff from like a year ago here?

PS FOund it: https://fractalforums.org/fractal-mathematics-and-new-theories/28/julia-and-parameter-space-images-of-polynomials/2786/msg14236#msg14236


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