Variability of Local Convergence and Divergence Rates Over the Attractor

Does the slope b vary with the two trajectories' location on the attractor? (Because a trajectory visits successive locations as time progresses, examining different locations on the attractor is tantamount to studying effects over time.) Continuing with the logistic equation, we'll fix k at 2.8 and conduct five separate experiments, corresponding to reference-trajectory starting values x0 of 0.05, 0.20, 0.50, 0.80, and 0.95, respectively. In each case, we'll set δ₀ as 0.01. Starting values of the nearby trajectories therefore become 0.06, 0.21, 0.51, 0.81, and 0.96, respectively. For all five experiments at k=2.8, the rate of convergence (slope b) turns out to be the same, namely -0.22 (the plots aren't shown here). Therefore, the local rate of convergence seems to be the same, at least for k=2.8 (a nonchaotic case).
Neighboring trajectories on chaotic attractors, on the other hand, don't necessarily behave so consistently. Slopes for the Lorenz attractor, for example, range from -15 to +16, depending on the particular location on the attractor and other factors (Nese 1989). Two adjacent trajectories along the tops of the attractor's lobes can even converge, producing negative values of the slope b for that small zone. In contrast, adjacent trajectories monitored at the bottoms of the two lobes separate rapidly, yielding large and positive values of b. Therefore, we'll have to start referring to a "local slope" (a value of b that pertains to a particular local region on an attractor) from now on.

The divergence or convergence of neighboring trajectories at different rates depending on phase space location complicates chaos analyses. Nicolis et al. (1983) proposed to account for that variability with a non-uniformity factor (NUF). The NUF treats local slope as a statistical variable because it varies from place to place on a chaotic attractor. The many different values can be compiled into a frequency distribution. The Nicolis et al. NUF is simply the standard deviation of that distribution of local-slope values. Thus, the NUF is a global indicator reflecting the spread of the local slopes about the mean local slope.