Extracting persistent topological modes of variability in complex dynamics from data

Complex systems such as the climate are often described in terms of linear modes of variability, but these modes cannot capture the intrinsically nonlinear organization of the dynamics. We introduce a framework for extracting topological modes of variability (TMVs) directly from observational, laboratory or simulation data. 

TMVs were introduced in the context of the templex framework [Charó et al., 2022; 2025], which represents a dynamical system through a combination of its topological structure and the way the flow in phase space moves across it. In this framework, TMVs correspond to flow patterns that are organized around special regions of an attractor, called joining loci, where different pathways merge.

Here we show how these joining loci — and the TMVs organized around them — can be recovered directly from data, without explicitly constructing a cell complex. We use dynamical indicators of local dimension and stability [Lucarini et al., 2016; Faranda et al., 2017] to locate the regions of the attractor where joining loci are expected, and we then extract the corresponding cycles from a directed graph built on a clustering of the data. By retaining only the robust transitions in this graph, we obtain a set of persistent TMVs.

We apply this approach to the El Niño–Southern Oscillation (ENSO) using Niño-3.4 sea-surface temperature anomalies from NOAA’s Oceanic Niño Index (ONI), providing new insight into ENSO variability and predictability.