2026/04/01 by Bernd von Mallinckrodt · 1 voice
Environmental Science · Physics and Astronomy · #Chaos control and synchronization #Ecosystem dynamics and resilience #stochastic dynamics and bifurcation
paper · doi:10.5281/zenodo.19367217
openalex publication_date 2026/04/01 · openalex created_date 2026/04/02 · openalex updated_date 2026/07/01
This work presents a conditional dual-metric framework for analyzing stability loss in multivariate dynamical systems approaching critical transitions. Classical early warning signals based on critical slowing down (e.g., variance and lag-1 autocorrelation) are theoretically grounded for low-dimensional systems with uniform slowing, but may fail to capture structural changes in high-dimensional systems. The proposed framework separates two distinct processes: (i) the geometric organization of state-space fluctuations, and (ii) the system’s responsiveness to external perturbations. Structural properties are quantified by the effective rank of the empirical covariance matrix, expressed as the exponential Shannon entropy of its normalized eigenvalue spectrum. This quantity captures spectral concentration and is shown to be non-redundant with scalar indicators under asymmetric mode compression. Dynamic responsiveness is defined via the H₂ norm of an empirically estimated input–output transfer function, measuring the total energy of system response across frequencies. A composite indicator, defined as the ratio of responsiveness to structural dimensionality, is introduced under a Structural–Dynamic Separability (SDS) condition requiring independent estimation and approximate causal independence of the two components. Analytical results derived for multivariate Ornstein–Uhlenbeck systems demonstrate that the composite indicator does not exhibit a universal monotonic trend near bifurcation. Instead, its behavior depends on the relative rates of structural compression and responsiveness change, as well as on the alignment between external inputs and the system’s critical dynamical mode. The framework is explicitly restricted to systems near fold-type bifurcations with identifiable external inputs and excludes noise-induced, rate-induced, and Hopf-type transitions from its domain of validity. No empirical performance claims are made. A pre-registered falsification protocol is provided to enable rigorous future testing. critical transitions early warning signals multivariate dynamical systems spectral entropy effective rank covariance structure input–output systems H2 norm Ornstein–Uhlenbeck process system stability high-dimensional systems regime shifts