Local Lyapunov Exponents

The exponent b—the exponential rate of trajectory convergence or divergence as derived for any one local region on an attractor—is a so-called local Lyapunov exponent. It characterizes a system's dynamics for that particular region, that is, over the associated brief timeframe. Because those local dynamics and hence expo nents can vary with location on the attractor, it's important always to attach the adjective ''local" when referring to such exponents. They aren't the same as the standard Lyapunov exponent that we're going to derive in the next section. A few representative papers dealing with local Lyapunov exponents include Fujisaka (1983), Nicolis et al. (1983), Grassberger & Procaccia (1984), Nese (1989), and Abarbanel (1992); see also Abarbanel et al. (1991a) and Abarbanel et al. (1992).
Local Lyapunov exponents can suggest attractor regions of potentially greater short-term predictability. For one thing, the duration over which the exponential relation holds tells us something about the time limits on predictability. Also, the value of the exponent reveals something about the reliability of predictions. For instance, a large and positive local Lyapunov exponent (fast divergence of neighboring orbits) suggests great sensitivity to initial conditions and poor predictability. Negative local exponents, in contrast, suggest convergence of neighboring trajectories and hence relatively good predictability. Measuring—and separately evaluating—local exponents over the entire attractor can tell us how they compare to the global average, their variability (which can be large) over the attractor, the special conditions for which predictions are reliable or unreliable, and many other things.