2026/04/14 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.19568423
openalex publication_date 2026/04/14 · openalex created_date 2026/04/15 · openalex updated_date 2026/07/01
Structural Compression Precedes Variance Amplification Near Fold Bifurcations This work analyzes the behavior of multivariate stochastic systems approaching fold (saddle-node) bifurcations through the geometry of their stationary covariance structure. While classical early warning signals focus on variance amplification and critical slowing down, these indicators primarily capture late-stage dynamics close to the critical point. We introduce a structural perspective based on the spectral entropy of the covariance matrix and its normalized effective rank (Φnorm). Using a linearized Itô system and the Lyapunov equation, we show that covariance eigenvalue concentration leads to a progressive reduction of effective dimensionality as the dominant mode diverges. This process—referred to as structural compression—is observable across the full pre-bifurcation regime. Analytical results demonstrate that Φnorm decreases monotonically toward zero under standard assumptions (spectral gap, non-degenerate noise, near-normal dynamics), with asymptotic scaling proportional to |α₁|·|log|α₁||. Numerical simulations of a 4-dimensional Ornstein–Uhlenbeck system confirm that structural compression unfolds gradually, while total variance remains nearly constant until the final stage before the bifurcation. The key empirical finding is not an earlier threshold crossing in a statistical sense, but a redistribution of informative signal across the control parameter: Φnorm utilizes its dynamic range continuously throughout the approach, whereas variance concentrates its change near criticality. We interpret Φnorm as a quasi-order parameter describing the geometric collapse of covariance structure. A complementary recovery proxy (AR(1) coefficient of the leading principal component) provides an auxiliary dynamical channel, enabling a joint structural–temporal diagnostic framework. The results establish structural compression as a robust and interpretable mechanism underlying early warning behavior in multivariate systems, with potential applications in ecology, climate dynamics, financial systems, and other complex adaptive domains. 🔬 Core scientific early warning signals critical transitions fold bifurcation saddle-node bifurcation critical slowing down 🧠 Method / novelty structural compression spectral entropy effective rank covariance analysis multivariate systems 📊 Mathematical / physical Lyapunov equation Ornstein-Uhlenbeck process stochastic dynamics eigenvalue spectrum dimensionality reduction 🌍 Application fields complex systems system stability tipping points resilience early warning signals, critical transitions, fold bifurcation, structural compression, spectral entropy, effective rank, covariance analysis, Lyapunov equation, stochastic systems, Ornstein-Uhlenbeck process, tipping points, system stability