Page:Lawhead columbia 0054D 12326.pdf/180

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no probabilistic elements, then the system will be deterministic. Many (most?) chaotic systems of scientific interest are deterministic. The confusion here stems from the observation that the behavior of systems in chaotic regions of their state space can be difficult to predict over significant time-scales, but this is not at all the same as their being non-deterministic. Rather, it just means that the more unsure I am about the system’s exact initial position in state space, the more unsure I am about where it will end up after some time has gone by. The behavior of systems in chaotic regions of their state space can be difficult to forecast in virtue of uncertainty about whether things started out in exactly one or another condition, but that (again) does not make them indeterministic. Again, we will return to this in much greater detail in Section 3 once we are in a position to synthesize our discussions of chaos and path-dependence.

Exactly how hard is it to predict the behavior of a system once it finds its way into a chaotic region? It’s difficult to answer that question in any general way, and saying anything precise is going to require that we at least dip our toes into the basics of the mathematics behind chaotic behavior. We’ve seen that state space trajectories in chaotic region diverge from one another, but we’ve said nothing at all about how quickly that divergence happens. As you might expect, this is a feature that varies from system to system: not all chaotic behavior is created equal. The rate of divergence between two trajectories is given by a particular number—the Lyapunov exponent—that varies from system to system (and from trajectory to trajectory within the system[1]). The distance between two trajectories x0xt and y0yt at two different times can, for any

  1. Because of this variation—some pairs of trajectories may diverge more quickly than others—it is helpful to also define the maximal Lyapunov exponent (MLE) for the system. As the name suggests, this is just the largest Lyapunov exponent to be found in a particular system. Because the MLE represents, in a sense, the “worst-case” scenario for prediction, it is standard to play it safe and use the MLE whenever we need to make a general statement about the behavior of the system as a whole. In the discussion that follows, I am referring to the MLE unless otherwise specified.