a = 1.5, center at -0.153+2.163i

]]>Recall the formula, , and whatever you may remember about the sin() function. bugR() is periodic with period . bugR(z) = 0 for all , n an integer. For the even multiples, and for old multiples, . While these observations do not constitute a full proof, it does strongly suggest why the two shapes alternate along the real axis.

]]>The next few pictures are success out-zooms of Bugs #16 to get an idea of what the neighborhood looks like.

Today’s image is a zoom out by a factor of two.

For a quick recap, this formula is bugR, defined back in Bugs #14 , with parameter a = 2.5.

]]>With z^2+c and a little algebra, you can show that if |z| > 2 + |c| then z escapes. Yesterday I showed a similar argument for bugI() which concludes that if |x| > a + 1 + |c| then z =x + yi escapes. So for both of these formulas, there are easily identified large areas where the orbit is guaranteed to escape. Also we saw that this type of simple algebra argument breaks down for bugR(). That does not prove that no such area exists, only that if it does, it will be difficult to find. Let’s dig deeper

Since this is a blog, not a book, I should recap frequently. We are iterating

Recall that

If a>.5 (an easy assumption, it has been true for every picture so far), u(p)=-.5 has a solution. In fact many solutions when you find one, is also a solution for any integer n.

Look at . where n and m are arbitrary integers.

These points lie on a grid. Equal space horizontally, decreasing spacing vertically because of the square root. But extending to infinity in all directions. Where ever you are in the plane you are surrounded by these points. The following will show that none of these points escape, in fact every one is a 2-cycle.

Lets iterate bugR(z^2+c)

So every one of these points generates a two cycle.

It is embarrassing to admit how much time I spent on this. My math skills are rusty.

]]>To save typing and simplify some calculations, define . Note that .

With this notation, . Set and expand the iteration .

After the first step, we always have . With a little algebra, it is seen that if . So the real part of the iteration keeps getting larger, and we have many orbits escaping to infinity.

A similar analysis fails with . On each iteration step,

The real part is always bounded, , so the only hope of escape lies with the imaginary part. But if x is a multiple of a*pi, then u(x) = 0, and y has to start over.

]]>Compare this image with Bugs #8. Same location, zoom, and color, same parameter a = 2.5, but with the R version replacing the I version of bug. The webbing appears to be everywhere, although it quickly becomes too thin to see at this image resolution.

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