Wednesday, March 19, 2008

Chaos and the problem of long periods

And so, back to the strangely soothing subject of chaos. If it's happened before, it might happen again.

Nonlinear systems exhibit subharmonics of low frequency, so low that the associated behavior happens only occasionally. If we don't know the fundamental dynamics of a system and can't work out its deterministic logic from first principles, then we are left with staring its empirical behavior, over a necessarily finite time. If we see something happen, how do we know it's a unique event or if it will repeat? After all, its period might just be really long. Is the system not just nonlinear, but chaotic as well?

Stated in a more technical way, empirical study of chaotic systems in always limited by the longest observed time scale. The reciprocal of that time scale gives the lowest frequency we can know about. Without being able to observe the system for an infinitely long time means we can never know for sure what's a unique event and what just takes a long time to repeat.

There's no general solution to this problem, given the restrictions on our knowledge just stated. We might try to ascertain the system's fundamental dynamics and so try to answer the question through mathematical deduction. We might try to extend our observing period through paleo-knowledge, indirect use of old proxy evidence that allows us to infer something about the system over time scales longer than human history. Evolutionary biology and paleoclimate are reconstructed through such techniques. All of these play a large role in the study of chaos in real life.

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