Disentangling informative and non-informative dynamics between time signals in chaotic systems

We introduce a method to decompose a (source) time signal into its informative and non-informative components with respect to another (target) time signal. The decomposition is constructed such that the informative component captures all the information necessary to reconstruct the future states of the target signal, while the non-informative component shares no information with the target in the future. Relying on the concept of Shannon information, the decomposition enables the definition of two quantities: an informative energy ratio, which measures the information content of the source variable relative to the target, and a sensitivity map, which characterizes the propagation of uncertainty from source to target. We demonstrate the applicability of the proposed decomposition in three scenarios. First, using the Lorenz system, we show that the decomposition can identify regions in phase space with low and high uncertainty for temporal predictions. Second, we apply the method to a one-way coupled Lorenz-Lorenz system to uncover the underlying coupling term. Finally, we employ the decomposition in a high-dimensional, chaotic setting – turbulent channel flow – to investigate the origins of drag fluctuations. Additionally, we demonstrate that forecasting models rely on the informative component extracted by the decomposition.