What are the practical applications of sets converging to repeating values?

[ktoz]

Back in the fractal craze, I wrote a simple application to generate the Mandelbrot set, and after way too many wasted hours, I noticed that the generating function frequently converged to sets of repeating values rather than single values. For example, for a 5 value convergent, the terms of the set are related by:

f(x1) = f(x0)
f(x2) = f(x1)
f(x4) = f(x3)
f(x0) = f(x4)

I have two questions related to this:
- Do sets of values that are related by these types of loops, have a name?
- Do these types of convergents have any practical applications?

Reason I ask is that I'm playing around with ideas for a "loopless" computer language and have come up with a few formulas that can eliminate iteration in specialized cases but these "poly-convergents" have always interested me as a potential way to directly calculate more complex states. Problem is though, I don't know what they're called.

Thanks for any info

 

[Hurkyl]

It's called an "attracting cycle".

As you adjust the parameters, you can watch the cycle "bifuricate"; e.g. you can watch a fixed point split into a two-cycle.

 

[ktoz]

It's called an "attracting cycle".

Thanks Hurky