2026/04/16 by Bernd von Mallinckrodt · 1 voice
Environmental Science · Physics and Astronomy · #Chaos control and synchronization #Complex Systems and Dynamics #Ecosystem dynamics and resilience
paper · doi:10.5281/zenodo.19604035
openalex publication_date 2026/04/16 · openalex created_date 2026/04/17 · openalex updated_date 2026/07/01
This paper introduces the Compression–Response Transition Index (CRTI), a composite early warning indicator designed to detect proximity to fold (saddle-node) bifurcations in multivariate systems. CRTI integrates two complementary components of pre-transition dynamics: (i) structural compression, quantified via the spectral entropy–based effective rank of the covariance matrix (Φ), capturing the redistribution of variance across system modes; and (ii) recovery dynamics, estimated from the AR(1) coefficient of the leading principal component (R), reflecting critical slowing down in the dominant mode. The index is defined as T(t) = R(t) / Φ(t) and is interpretable only under a formally defined Structural–Dynamic Separability (SDS) condition, which ensures that structural and dynamical signals remain sufficiently independent. A boundary condition, the Relaxation–Coupling Failure Mode (RCFM), is explicitly characterised, under which CRTI loses interpretability and does not outperform classical univariate early warning signals. The theoretical motivation is grounded in the Lyapunov equation relationship between the Jacobian and the covariance matrix, which predicts directional variance concentration as a system approaches a fold bifurcation. A minimal numerical demonstration illustrates the qualitative behaviour of Φ, R, and CRTI near a transition. CRTI is explicitly scoped to multivariate systems (d ≥ 2) with approximately isotropic noise and does not apply to Hopf bifurcations, noise-induced transitions, or rate-induced tipping. The framework is presented as a mechanism-specific diagnostic tool intended to complement, not replace, classical early warning signals. This paper serves as the primary reference and entry point for the CRTI framework. core critical transitions early warning signals fold bifurcation structural compression covariance dynamics CRTI-spezifisch: spectral entropy effective rank multivariate systems principal component analysis Positionierung: complex systems nonlinear dynamics tipping points Keywords: critical transitions; early warning signals; fold bifurcation; structural compression; covariance dynamics; spectral entropy; effective rank; multivariate systems; principal component analysis; nonlinear dynamics; tipping points; complex systems