Convergent vs Divergent

From Ultrafractal Wiki

Ed wrote: "how can i tell what 'type' (i.e. Convergent or Divergent) a fractal is? and what is the difference?"

With the typical Mandelbrot or Julia fractal, the outside points are colored and the inside points are solid black (obviously, any particular image could be rendered in a variety of ways). These are both divergent fractals. With divergent fractals, the outside points (the points that aren't iterated for the maximum number of times) have orbits (the sequence of z values) that head off to infinity. With convergent fractals, the outside points have orbits that settle in on, or converge to, a specific set of values. An example is the typical Newton's Method fractal. However, in the early days, coloring methods were primarily written for divergent fractals, so some convergent types were written as divergent. Consequently, it's not easy to tell if a given formula will give you convergent or divergent unless you look at the code.

Kerry

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Hi,

One indicator as to whether a main formula is convergent or divergent is the bailout parameter for the formula.

If it's called "Bailout" and the default value is >1 (typically say 4 or 100 or 128 or 65536 or 1e20) then that's a divergent bailout so the formula is probably divergent.. If it's called say "Small Bailout" and the value is <1 (typically say 1e-5) then it's a convergent bailout so the formula is probably convergent Note that the default value being <1 or >1 is the definitive indicator (rather than the parameter name).

Some formulas are both divergent and convergent so have both types of bailout parameter - such as the standard Magnet formulas.

The divergent bailout values are used for testing if the orbit of z is tending to infinity - if the magnitude of z at any time in the orbit exceeds the bailout value then it's assumed that z is tending to infinity and the pixel is coloured as "outside". The convergent bailout values are used for testing if the orbit of z is tending to a fixed finite constant - it may be used in one of two ways depending on the main formula. If the fixed finite constant is known beforehand then if at any time in the orbit of z the magnitude of (z-fixed constant) is smaller than the small bailout value then the pixel is coloured as "outside". In formulas where the fixed constant isn't known beforehand then if at any time in the orbit of z the magnitude of (z-previous z) is smaller than the small bailout value then the pixel is coloured as "outside".

bye Dave

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Except as Kerry pointed out, some earlier formulas may have a large bailout even though they are convergent by doing some tricks with the programming. For example, the Newton formula in lkm.ufm is one such formula.

Ron Barnett

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Hi,

Just re-read what Kerry said - it's true that some (usually older) convergent fractal formulas are written in such a way that they act like divergent ones - in these cases the bailout parameter will be a divergent one and for colouring purposes you may need to set colouring parameters accordingly i.e. as if the formula was divergent - then again some such convergent formulas may only treat the bailout test in a divergent manner and in these cases you'd have to colour the fractal as a convergent one even though it's using divergent bailout.. Fortunately I don't think there are many such formulas - at least not in common use.

Also I should have added that in some formulas the bailout value used is hard-coded into the formula i.e. there will be no bailout parameter of either kind. In these cases you'd have to look at the formula code to see whether the formula is convergent, divergent or both and to find out the bailout value/s that's used. Again the bailout value used will (normally) tell you whether the formula is convergent or divergent.

bye Dave