Mandelbrot-Variant Orbit Traps

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Offline ixm.ibrahim

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« on: March 03, 2018, 01:19:44 AM »
I've been using C# to generate classic fractals, such as the Mandelbrot, Burning Ship, and their corresponding Julia Sets. For those kinds of fractals, the general orbit trap used is testing whether the orbit escapes a circle with radius of 2, as when the radius gets beyond 2 the coordinate we're iterating diverges.

My first question is what math was used to prove this? I've looked at some sources that explain this but the way they explain it doesn't help my understanding. Looking at Wikipedia for example, I'm at a loss to where the '2' in the formal definition comes from.


I want this information to help me figure out the conditions for divergence for other Mandelbrot functions, such as Sin(z) + c. To get decent images of that function I need to make the bailout value 20 or more, but then if I zoom out the fractal gets cut off very quickly. Just as a test, instead of testing for radius < 4, I changed it to Sin(radius) < 4, which yielded much better results while maintaining a bailout value of 2.


Any math or resources to help with these topics would be very much appreciated!

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/mandelbrot-variant-orbit-traps/964/

Offline claude

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« Reply #1 on: March 03, 2018, 05:45:57 PM »
For the quadratic z^2 + c, see https://math.stackexchange.com/questions/890190/mandelbrot-sets-and-radius-of-convergence/890448#890448

For general polynomials, see Lemma 14.1 in Falconer's "Fractal Geometry: Mathematical Foundations and Applications" (2nd ed):
Quote
Given a polynomial \( f(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_0 \) with \( a_n \not = 0 \), there exists a number r such that if |z| >= r then |f(z)| >= 2 |z|. In particular, if \( |f^m(z)| \ge r \) for some m >= 0 then \( f^k(z) \to \infty \text{ as } k \to \infty \). Thus either \( f^k(z) \to \infty \) or \( \left\{ f^k(z) : k = 0, 1, 2, \ldots \right\} \) is a bounded set.
The proof of the lemma doesn't say how to find the smallest such r, as far as I can understand it, but I suspect r may need to be bigger than 2 in some cases.

For other functions, escape radius in the form of a circle may not be meaningful.  For f(z) = sin(z) + c maybe try testing just the imaginary part?  Not sure..


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