Stability Window

Robert P. Munafo, 2023 Jun 23.

For an iterated function exhibiting suitable complexity (see period 3 implies chaos), a "stability window" is a region in the parameter plane within which iteration is stable in the sense of having large contiguous regions within which all parameters have the same finite period. These "windows" always include a period doubling cascade and adjacent regions of "chaos" (points with non-periodic orbits) notably also including smaller stability windows whose periodicity is an integer multiple of the larger window.

Put in terms of the Mandelbrot set, the above paragraph states:

For the Mandelbrot iteration function, which includes the period-3 mu-atom R2F(1/2B1)Sa, a "stability window" is any island mu-molecule, which is a region of the parameter plane within which iteration is stable because the island is of nonzero size and contains mu-atoms that each have a finite period. These islands always include a period doubling cascade (their cardioid and the .1/2a, .1/2.1/2a, .1/2.1/2.1/2a, etc. mu-atoms connected to it) and adjacent filaments containing non-periodic points, but notably also including smaller islands whose period is an integer multiple of the period of the larger island.

Estimating the Size and Orientation of an Island

If an island's period and nucleus coordinates are known, it is easy to estimate the "size" of the island and the direction it is "pointing".

The mathematical principles involved are periodicity scaling, renormalization, and tangent bifurcation; they combine to allow a size estimate as explained here by Evgeny Demidov.

Given the nucleus coordinate c and the period p of an island, the basic principle is to accumulate (via a series product) the iterates of c through p-1 iterations. This is similar to the normal derivative calculation but omits the "+1" at each step, because it is iterating in the Julia plane. In addition, the iteration is done for one fewer steps because the derivative is very small at the critical point (f^p(c)=0).

Λ = 1
β = 1
z = 0
repeat the following p-1 times:
       z = z²+c
       Λ = 2 z Λ
       β = β + 1/Λ
the size estimate is:
v = 9/(4 β Λ²)

As you can see, Λ ends up being 2^p-1 times the product of the first p-1 iterates. The final v is the "size estimate", but since it is a complex number it has a magnitude and angle. The magnitude is nearly equal to the distance from the island's .C(0) cusp to its t tip, and the angle tells which direction the island is "pointing".

There is a constant multiplier 9/4 in the v calculation, and this is precisely the distance from the main cusp R2.C(0) to the main tip R2t. Thus, if we leave off the "9/4" part of the formula we get a scaling "ratio" telling how small the island of interest is in comparison to the whole Mandelbrot set.

revisions: 20230620 first version; 20230623 corrections

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