Testing for Partially Predictable Chaos in Delayed Dynamical Systems

For deterministic chaos initially close trajectories diverge exponentially leading to a loss of correlation followed by a diffusive decorrelation. If the decorrelation of trajectories is dominated by the diffusive process happening on a much longer time scale than the Lyapunov prediction time, the chaotic attractor is partially predictable for long time [1]. However, due to the high level of correlation, the standard tests for chaos either yield ambiguous results or misclassify partially predictable chaos as laminar flow.
In this study we present a novel test for chaos based on the scaling of the long-term distance of trajectories. The test distinguishes chaos and laminar flow robustly in a 0-1 fashion, including partially predictable chaos. The correlation of trajectories serves as another binary measure to discriminate partially predictable chaos and ordinary chaos.
The combination of both tests is used to classify each of the three states - chaos, partially predictable chaos and laminar flow - by computing pairs of trajectories. Partially predictability chaos can be observed in autonomous dynamical systems but is also found in delayed dynamical systems being formally of infinite dimension.

[1] Wernecke, Sándor, Gros, How to test for partially predictable chaos. SciRep 7(2017).