How to test for partially predictable chaos

Abstract: For a chaotic system pairs of initially close-by trajectories become eventually fully uncorrelated on the attracting set. This process of decorrelation may split into an initial exponential decrease, characterized by the maximal Lyapunov exponent, and a subsequent diffusive process on the chaotic attractor causing the final loss of predictability. The time scales of both processes can be either of the same or of very different orders of magnitude. In the latter case the two trajectories linger within a finite but small distance (with respect to the overall extent of the attractor) for exceedingly long times and therefore remain partially predictable. Tests for distinguishing chaos from laminar flow widely use the time evolution of inter-orbital correlations as an indicator. Standard tests however yield mostly ambiguous results when it comes to distinguish partially predictable chaos and laminar flow, which are characterized respectively by attractors of fractally broadened braids and limit cycles. For a resolution we introduce a novel 0-1 indicator for chaos based on the cross-distance scaling of pairs of initially close trajectories, showing that this test robustly discriminates chaos, including partially predictable chaos, from laminar flow. One can use furthermore the finite time cross-correlation of pairs of initially close trajectories to distinguish, for a complete classification, also between strong and partially predictable chaos. We are thus able to identify laminar flow as well as strong and partially predictable chaos in a 0-1 manner solely from the properties of pairs of trajectories.