Umpteen ways to look at chaos

There are at least three distinct and powerful ways to look at chaos. So far, we've been looking at only one: the aperiodicity path. The other two are the sensitivity-to-initial-conditions ("butterfly effect") path and the statistical path. All three are more or less equivalent, but have greater or lesser degrees of flexibility and generality.

The common way chaos has been discussed, both by scientists and in popular culture, goes by the poetic name of the "butterfly effect," otherwise known technically as "exponential sensitivity to initial conditions." The idea is that a system whose dynamical rules (the rules by which it changes in time) are known is deterministic: if the system's state is known exactly at some time, it can be known exactly for all time, at least in principle.

The catch is that no real system's state is ever known exactly in real life. So determinism in practice means the following, by no means obvious, requirement: approximately knowing the system state means being able to approximately predict its future or reconstruct its past. Chaotic systems severely curtail this possibility, because a small error in knowing the system's state at one time gets blown up exponentially fast into a much larger uncertainty. The error ε(t) a time t later after starting is ε(t) ~ ε(0)·exp(λt). λ is the Lyapunov exponent. It has units of inverse time; its reciprocal 1/λ is the Lyapunov time. In that time after starting, roughly, you can no longer predict the system's behavior. The presence or absence of a butterfly's flapping wings in Brazil could lead or not lead to a hurricane in Florida, or so the saying goes. No mathematical-computational model mimicking the system can compute faster or better than the system itself.

A chaotic system is its own best computer.

Determinism is a powerful concept and fundamental to scientific knowledge. I know the system exactly at an instant. What happens next is fixed. Determinism differs from randomness or stochasticity, where what happens next is one of a set of possibilities, each with a probability attached. Chaos shows how an ordered, deterministic system can look random, and how apparent randomness is really a fantastic sort of order.

While this approach to chaos is valid, it's not as useful as the other two approaches. It requires that you know the system's complete dynamics (even if you can't solve it). If you don't, the frequency-event-Fourier approach, or the statistical way are much more helpful. These other two methods are useful for nonlinear, complex systems whose dynamics is too hard. In the case of weather, the basic equations are known, but are far too complicated to solve. In the case of biological evolution or the financial markets, no one knows more than a fragment of the dynamics in an exact mathematical form.

A statistical look at chaos. To analyze chaos statistically, forget about evolution in time. Just bin "changes" in the system by how often they occur, by size.* Some very interesting distributions emerge, power laws, not the Gaussian bell curve or one of its cousins (say, the Poisson or binomial distributions). Large changes are less probable than smaller ones, but their probability declines slowly, as a low power of the size, not at all the rapid falloff of the Gaussian. This phenomenon sometimes goes by the name of "fat" or "long tails."

The two alternative approaches can be applied in these cases, where the other one fails. As long as degrees of freedom and subsystems can be identified, and a few, quite general assumptions are made, chaos in many systems can be investigated with these methods. They're agnostic about what the fundamental dynamics is. They might even help to "reverse engineer" or reconstruct the dynamics when it is not initially known.**
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* This is clearly related to the Fourier approach, just interchanging the axes. The "size" of change is related to the "power" (the latter is essentially the square of the former).

** In technical jargon, this is called "reconstructing the phase space" - we'll meet the concept again.