Spatio-temporal dynamical indices for complex systems

Complex systems span multiple spatial and temporal scales, making their dynamics challenging to understand and predict. This challenge is especially daunting when one wants to study localized and/or rare events. Advances in dynamical systems theory, including the development of state-dependent dynamical indices, namely local dimension and persistence, have provided powerful tools for studying these phenomena. However, existing applications of such indices rely on considering a predefined and fixed spatial domain, which provides a single scalar quantity for the entire region of interest. This aspect prevents understanding the spatially localized dynamical behavior of the system. In this work, we introduce Spatio-temporal Dynamical Indices (SDIs), which leverage the existing framework of state-dependent local dimension and persistence. SDIs are obtained via a sliding window approach, enabling the exploration of space-dependent properties in spatio-temporal data. As an example, we show that we are able to reconcile previously different perspectives on European summertime heatwaves. This result showcases the importance of accounting for spatial scales when performing scale-dependent dynamical analyses.