(Question) Precision in Lyapunov images?

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Offline marcm200

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« on: January 30, 2019, 02:16:22 PM »
pauldelbrot uses arbitrary precision in his wheel 32 image and sees detail at great depths. As for the Lyapunov images, I use the C++ double floating point type and see not much of a difference in iteration depths, neither does ThunderblotPagoda with his single type.

But usually I look at ranges in the plane not smaller than 10-2. Let's say r is from 5,01 to 5,02. So using a function like sin(x+r) and x being between -1 and 1, there is no problem adding these two numbers in C++ as they are roughly the same order of magnitude. However when I zoom in and the r-value gets like 5+10-50, adding this value to an x of 0.5 will always result in 5.5. I guess that's why I'm always getting images that become emptier and emptier as I zoom in.

I was wonderung: Has anyone done calculations for the Lyapunov exponent with arbitrary (or very high precision) so adding two values of a largely different magnitude was feasible? Do those images show detail or are the Lyapunov images an empty desert after some specific point of zooming?

Linkback: https://fractalforums.org/programming/11/precision-in-lyapunov-images/2578/

Online claude

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« Reply #1 on: January 30, 2019, 04:31:30 PM »
5±10-50 is already equal to 5 in double precision, if you plot the r plane near 5 you could go as small as maybe 5±10-14 depending on pixel count, any deeper and you'll get increasing amounts of pixelation as neighbouring pixels have the same r value.  Single precision float limit would be around 5±10-5.

Regarding detail, it depends on the formula and where you look.  For x→x^2+c (conjugate to x→rx(1-x)), there are likely stable "minishapes" arbitrarily near -2 and other Misiurewicz points, though they probably get very small very quickly.  The minishapes are similar to the whole, so they will have their own minishapes surrounding them.

To go deeper, arbitrary precision is relatively easy to implement, but perhaps perturbation techniques could be applied, similarly to the Mandelbrot set?

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