Rossler Attractor
[Rossler]


Rossler's attractor is not a famous attractor, but is a rather nice attractor which draws a nifty picture. The attractor is formed with another bunch of Navier-Stokes equations, namely:

dx/dt = -y - z

dy/dt = x + Ay

dz/dt = B + xz - Cz
A, B, and C are constants.

The unique part of this attractor is that it displays banding. To understand banding, here is a little thought experiment, again. Start with the Cantor set. The Cantor set is simple to create; take a line, and trisect it; then cut out the middle third. You are left with two lines. Do the same for both lines; trisect them, and punch out the middle of those. With the four lines remaining, do the same thing. Now iterate this method forever. One will then have the "Cantor set," or "Cantor dust." This is an infinite number of infinitely small points arranged in a definite pattern.



Take the last iteration possible (the infinite one--just assume here that iteration five is infinite. I know it's not, but who's counting? Even my SuperVGA is incapable of showing that kind of infinitely small resolution). Find and mark the midpoint.

Rotate the iteration around the midpoint, and you will get what is known as the Cantor target. This is an infinite number of concentric circles, arranged so that if you took a diameter-slice out of it you would end up with the Cantor set dust. The arrangement of the circles is known as "banding."

Rossler's attractor displays a type of banding, which suggests that perhaps it is related to the Cantor set. Another interesting fact about Rossler's attractor is that it has a half-twist in it, which makes it look somewhat like a Möbius strip (what you get when you take a strip of paper, half-twist it once, and tape the ends together. A trick that almost everybody knows is to cut along the middle of the band, you will end up with a double-loop; and if you cut in the middle of that double-loop, you will end up with two separate, linked rings).

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