Monday, March 31, 2008

In the face of chaos or, Chaos casts a shadow

Chaos is the score upon which reality is written.
- Henry Miller

Any natural system, unless proven otherwise, should be assumed to be nonlinear and possibly chaotic. If we can prove it's not linear, we then need to ask, is it chaotic? Under what conditions? If not, under what conditions? Then to move on: can basic degrees of freedom and distinct subsystems be identified? Can their trajectories in time be plotted?

The weather, the state of the Earth's atmosphere, is known to be chaotic, with a Lyapunov time of about two weeks. That is, given anything like our current knowledge of the Earth's weather at any instant of time, weather predictions can be projected out to no more than two weeks, before the forecasts become no better than random guesses.

Financial markets are also known to be chaotic, but unlike the weather, we have no precise, laboratory-controlled fundamental principles to start with. There are many different kinds of financial markets as well. They do seem to lose any distinctive predictability after periods ranging from months to a few years.

What to do? We can't make long-term predictions of chaotic behavior. But chaos is bounded aperiodicity, not unbounded. Over time, it traces out increasingly and ultimately infinitely complex trajectories of temperature, pressure, precipitation, or prices and commodity flows. They're bounded, however, which suggests the notion of a "box" or a "range."

Mathematicians have given us a more subtle and precise version of a "chaos box," called a stable manifold.* Once the periodic and transient behaviors of a system are analyzed and removed from consideration, what's left is the untameable but still boundable meanderings of chaotic motion as it traces out an attractor. The actual record of chaotic trajectories is an infinitely complex fractal, but a stable manifold shadows that fractal in such a way that the manifold does not change over time and the chaotic trajectories don't cross it.**

Such stable manifolds are "chaos captured," to the extent that it can be. They give us a way to take a for-all-time snapshot of chaos without attempting the impossible task of tracing the actual chaotic trajectory for an infinite period.

An earlier posting linked to the stable manifold for the Lorenz attractor, the first modern model of chaotic turbulence arising in the atmosphere. The stable manifold shadows the Lorenz attractor and, being two dimensional, can be converted into a crochet pattern.
* "Manifold" is mathematics jargon for a space, like a two-dimensional surface embedded in three-dimensional space, that looks Euclidean locally, but maybe not globally. "Euclidean" means its geometry is the one you learned in high school.

** Mathematicians and physicists say that this object (the stable manifold) is invariant.

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