2026/04/15 by Bernd von Mallinckrodt · 1 voice
Economics, Econometrics and Finance · Environmental Science · Physics and Astronomy · #Chaos control and synchronization #Complex Systems and Time Series Analysis #Ecosystem dynamics and resilience
paper · doi:10.5281/zenodo.19594608
openalex publication_date 2026/04/15 · openalex created_date 2026/04/16 · openalex updated_date 2026/07/01
This preprint investigates spectral compression in multivariate systems as a structural diagnostic for critical transitions, focusing on the effective rank Φ of the covariance matrix as a measure of variance distribution across eigenmodes rather than total variance magnitude. We show analytically that in Ornstein–Uhlenbeck systems approaching a fold bifurcation via a single critical mode under isotropic noise, the subordinate eigenvalues remain constant while the leading eigenvalue diverges. In this regime, Φ collapses to a deterministic function of the leading eigenvalue fraction p₁ = λ₁/∑λᵢ, implying that Φ carries no independent information beyond p₁. This rank-1 collapse is not a limitation of the metric, but a structural property of the underlying dynamics. We derive the necessary conditions under which Φ can provide independent information: sufficient dimensionality (n ≥ 5), non-rigid subordinate spectra reflecting evolving coupling structure, and adequate sample size (T/n ≥ 20) to control finite-sample eigenvalue bias. Outside these conditions, Φ is expected to be redundant with simpler spectral summaries. The contribution of this work is twofold: (i) a formal characterization of the redundancy regime of effective rank under single-mode criticality, and (ii) an explicit criterion for detecting when multivariate covariance structure encodes information beyond variance and leading-mode dominance. Negative results in empirical settings are interpreted as diagnostic evidence for rank-1 dynamics rather than failure of the method. The results position effective rank not as a universal early warning signal, but as a conditional structural indicator whose utility depends on the geometry of the covariance spectrum and the dynamics of mode coupling. This reframing provides a principled basis for distinguishing between single-mode critical slowing down and higher-order spectral redistribution in complex systems. Core Keywords (wissenschaftlich präzise): spectral compression effective rank covariance eigenvalues fold bifurcation critical transitions multivariate early warning signals Mechanism / Theory: rank-1 collapse eigenvalue spectrum Ornstein–Uhlenbeck process critical slowing down spectral entropy Discoverability / breiter Kontext: complex systems nonlinear dynamics tipping points system stability high-dimensional systems