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.Comment: 14 pages, 9 figure
