Abrupt transitions ("tipping") in nonlinear dynamical systems are often accompanied by changes in the geometry of the attracting set, but quantifying such changes from partial and noisy observations in high-dimensional systems remains challenging. We address this problem with a sequential diagnostic framework, Data Assimilation-High dimensional Attractor's Structural Complexity (DA-HASC). First, this method reconstructs system's high-dimensional state using data assimilation from limited and noisy observations. Second, we quantify a structural complexity of the high-dimensional system dynamics from the reconstructed state by manifold learning. Third, we capture underlying changes in the system by splitting the reconstructed timeseries into sliding windows and analyzing the changes in the temporally local attractor's structural complexity. The structural information is provided as graph Laplacian and measured by Von Neumann entropy in this framework. We evaluate DA-HASC on both synthetic and real-world datasets and demonstrate that it can detect tipping under high-dimensionality and imperfect system knowledge. We further discuss how this framework behaves across different tipping mechanisms.
Abrupt transition points ("tipping points"), critical thresholds beyond which a system experiences a rapid and often irreversible shift, are a central theme in nonlinear dynamics and complex systems. In climate science, tipping has been highlighted in the context of climate change over the last two decades (van Nes et al., 2016), with widely discussed examples including melting of Greenland Icesheet, dieback of Amazon Rainforest, collapse of Atlantic Meridional Overturning Circulation (AMOC) etc. (Lenton et al., 2008). It is of paramount importance to develop mathematical methods to realize an improved understanding of these tipping phenomena and apply them to construct better Early Warning Signals (EWS).
Most of the widely used early warnings are rooted in critical slowing down (CSD) and are primarily designed and used for systems that gradually approach a bifurcation, i.e., bifurcationinduced tipping (B-tipping) (Scheffer et al., 2009;Boers, 2021). Two limitations are particularly explicit (Dakos et al., 2024). First, generic CSD-based warnings are not specific to abrupt and irreversible transitions; they can also respond to smooth and reversible changes, which induces false positives. Second, many early-warning tools are well tailored to unidimensional observables (Dakos et al., 2008), whereas most real systems are high-dimensional. The researches have thus increasingly emphasized approaches that treat multivariate dynamics explicitly via evolving networks analysis (Lu et al., 2021;Zhang et al., 2024) or dimension reduction by Principal Component Analysis (PCA) also known as Empirical Orthogonal Function (EOF) in climatology (Held & Kleinen, 2004), or to combine generic and system-specific information in a principled way (Flores et al., 2024).
However, these multivariate and hybrid directions introduce practical trade-offs. Network-based early warnings can be highly sensitive to how the network is defined, to thresholding choices, and to the stability of inferred connections. Stabilizing such pipelines typically requires phenomenoninformed design choices. PCA/EOF based approaches face a closely related issue: the extracted dominant modes and their trends can vary across parameter choices and over time. Therefore, even with careful preprocessing, their consistent effectiveness may break down under nonlinear, regime-dependent responses manifested through interacting drivers and delayed adjustments without system-specific prior knowledge about the dynamics. While Machine-Learning approaches (ML) have been proposed as an alternative approach and can achieve high accuracy on benchmark datasets (Bury et al., 2023), they make this ‘dependency on design choices problem’ even more explicit. Their performance depends critically and essentially on the construction of the training corpus and its match to the target system (Dakos et al., 2024).
These challenges become even more acute when focusing on applying to real Earth-system datasets. In these settings, observational sparsity, strong noise, and changes in the observation system can substantially affect early-warning estimates. Data assimilation (DA) offers a principled way to address these issues by integrating observations into process-based models to produce consistent state estimates and uncertainty-aware ensembles (Carrassi et al., 2018), and existing datasets are usually reanalysis data after DA. In this context, DA should be treated as an explicit component of the tipping-detection framework rather than as a “pre-preprocessing”.
These considerations motivate a tipping indicator that is training-free, multivariate, and operationally robust under noisy, sparse and indirect observations. We therefore frame tipping analysis as a trajectory and geometry structure-aware monitoring problem, where tipping-relevant change is reflected in the observed or assimilated (high-dimensional) state space rather than univariate temporal statistics. This viewpoint is related to orbit-based diagnostics from nonlinear dynamics. Most popularly, the Largest Lyapunov Exponent (LLE) quantifies asymptotic dispersion of deterministic chaotic attractor orbits (Benettin et al., 1980). In stochastic or partially observed systems, Ensemble-Averaged Pairwise Distance (EAPD) function has been proposed as a practical surrogate (“Instantaneous” LLE), with recent studies indicating potential applicability to high-dimensional systems (Jánosi & Tél, 2024).
First, we use DA as a key component to reconstruct the underlying high-dimensional system from limited data. Second, instead of applying conventional dimension reduction methods, we adopt manifold-learning-based approach to extract structural information without cutting off important high-dimensional information. We quantify this structural complexity as entropy, interpreting its fluctuations as a signal of regime shifts. This indicator is thus designed to capture not only the complexity of stable phases before and after a tipping event,
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