2026/04/18 by Bernd von Mallinckrodt · 1 voice
Environmental Science · Physics and Astronomy · #Ecosystem dynamics and resilience #Chaos control and synchronization #Complex Systems and Dynamics
paper · doi:10.5281/zenodo.19645082
openalex publication_date 2026/04/18 · openalex created_date 2026/04/19 · openalex updated_date 2026/07/01
Early-warning signals (EWS) for critical transitions are commonly based on scalar statistics such as rolling variance and lag-1 autocorrelation (AR(1)), which capture critical slowing down in low-dimensional systems. However, many real-world systems exhibit non-trivial covariance structure that cannot be resolved by scalar indicators alone. This work introduces a controlled theoretical framework to determine when structural diagnostics — specifically spectral-entropy-based measures of effective rank — provide information beyond classical scalar EWS. We formulate a necessary and sufficient condition for structural observability: structural information is extractable if and only if the observation preserves multiple independent degrees of freedom with non-degenerate covariance contributions. The result is demonstrated through a minimal benchmark comparing two regimes under identical protocols: (i) a one-dimensional stochastic fold bifurcation, representing the structure-absent limit, and (ii) a two-dimensional anisotropic Ornstein–Uhlenbeck system with evolving covariance geometry. In the scalar regime, structural diagnostics are provably redundant and statistically indistinguishable from variance. In the multivariate regime, structural measures capture covariance anisotropisation and separate system states that scalar indicators cannot distinguish. The findings generalize beyond any specific diagnostic and establish a principled boundary for the applicability of spectral and information-theoretic early-warning signals. Rather than proposing a universally superior indicator, the work identifies the conditions under which structural diagnostics become informative, reframing early-warning detection as a problem of observability rather than indicator design. All simulations are fully reproducible (NumPy-based Euler–Maruyama integration, fixed SeedSequence initialization), and the accompanying code and data archive allow exact regeneration of all figures and results. Keywords early warning signals, critical transitions, critical slowing down, stochastic systems, Ornstein–Uhlenbeck process, saddle-node bifurcation, covariance structure, spectral entropy, effective rank, observability, multivariate time series, complexity science, nonlinear dynamics, tipping points, system stability