Quantifying local predictability in phase space

Abstract

Both the classical Lyapunov exponents and the quantities from which they are derived, the local divergence rates, characterize the behavior of initially adjacent trajectories in phase space. The exponents are the long-term averages of this divergence, while the local rates quantify the behavior of nearby trajectories as a function of time and phase space position. Thus, given a transient-free initial state, the local divergence rates provide estimates of short-term predictability along the resultant trajectory. We calculate local divergence rates for the chaotic attractor of the three-component Lorenz model, and we investigate both the temporal and phase-spatial variations in predictability. We show quantitatively that predictability varies considerably with time, but that there is phase-spatial organization to the variability. This underlying structure permits identification of regions in phase space of high and low predictability and, in some cases, it permits an estimate of predictability to be assigned to the predictability variable itself.