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). 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.
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.