state variables are connected to one another.

The important feature of Lorenz’s system for our discussion is this: the system exhibits chaotic behavior *only for some parameterizations*. That is, it’s possible to assign values to σ, ρ, and β such that the behavior of the system in some sense resembles that of the pendulum discussed above: similar initial conditions remain similar as the system evolves over time. This suggests that it isn’t always quite right to say that *systems* themselves are chaotic. It’s possible for some systems to have chaotic *regions* in their state spaces such that small differences in overall state not when the system is *initialized*, but rather when (and if) it enters the chaotic region are magnified over time. That is, it is possible for a system’s behavior to go from non-chaotic (where trajectories that are close together at one time *stay* close together) to chaotic (where trajectories that are close together at one time diverge)^{[1]}. Similarly, it is possible for systems to find their way *out* of chaotic behavior. Attempting to simply divide systems into chaotic and non-chaotic groups drastically over-simplifies things, and obscures the importance of finding *predictors* of chaos—signs that a system may be approaching a chaotic region of its state space before it actually gets there^{[2]}.

Another basic issue worth highlighting is that chaos has absolutely nothing to do with indeterminism: a chaotic system can be deterministic or stochastic, according to its underlying dynamics. If the differential equations defining the system’s path through its state space contain

- ↑ The Phillips curve in economics, which describes the relationship between inflation and unemployment, is a good real-world example of this. Trajectories through economic state space described by the Phillips curve can fall into chaotic regions under the right conditions, but there are also non-chaotic regions in the space.
- ↑ A number of authors have succeeded in identifying the appearance of a certain structure called a “period-doubling bifurcation” as one predictor of chaotic behavior, but it is unlikely that it is the only such indicator.

169