Time-lagged recurrence: A data-driven method to estimate the predictability of dynamical systems

Nonlinear dynamical systems are ubiquitous in nature and they are hard to forecast. Not only they may be sensitive to small perturbations in their initial conditions, but they are often composed of processes acting at multiple scales. Classical approaches based on the Lyapunov spectrum rely on the knowledge of the dynamic forward operator, or of a data-derived approximation of it. This operator is typically unknown, or the data are too noisy to derive its faithful representation. Here, we propose a data-driven approach to analyze the local predictability of dynamical systems. This method, based on the concept of recurrence, is closely linked to the well-established framework of local dynamical indices. When applied to both idealized systems and real-world datasets arising from large-scale atmospheric fields, our approach proves its effectiveness in estimating local predictability. Additionally, we discuss its relationship with other local dynamical indices, and how it reveals the scale-dependent nature of predictability. Furthermore, we explore its link to information theory, its extension that includes a weighting strategy, and its real-time application. We believe these aspects collectively demonstrate its potential as a powerful diagnostic tool for complex systems.