Sounds kinda like grooves in a record. Very interesting. It reminded me of something I recently read, may've been here on Hubski, or links found via Hubski. That some basic building block of existence... photon? Gods, I can't remember. Anyhow, it travels faster than the speed of light, so in essence, it transcends time. All of time can be found in a single instance of that element, and at any given point in human time, it is existing in both earth's prehistorical periods and periods far into the future all at once.
Not so orderly as grooves on a record. Have you ever scribbled with a pen for a long time, making loop upon loop that all cross over a central point? Some loops are huge, others small, all eventually make their way back to the dark inky blotch in the middle. Something like that.
Sadly I don't remember much about them, but the Wikipedia article on attractors is actually pretty nice. Seems like attractors are just some set of numbers for which a system tends toward. Maybe the fixed-point attractor is a better descriptor of your original idea: the system tends toward one fixed point and oscillates around it chaotically. I first read about attractors in high school in the book Chaos by James Gleick, which is a fantastic book I highly recommend for this sort of stuff.