On the Asymptotic Equivalence of Fisher-Optimal and Eigenmode-Based Observation for Critical Transition Detection

This work analyzes the asymptotic relationship between Fisher-information-based observation design and classical eigenmode observability for early warning signals near critical transitions in stochastic dynamical systems. For finite-dimensional stochastic systems with additive noise near generic codimension-1 bifurcations, the paper proves a Rank-1 Cancellation Theorem showing that the Fisher information associated with scalar observations asymptotically collapses to the same observable subspace characterized by Popov–Belevitch–Hautus (PBH) eigenmode observability. Consequently, all scalar observations with nonzero projection onto the critical eigenmode become asymptotically equivalent in Fisher information, while blind directions remain uninformative. The manuscript further investigates potential escape routes — including multiplicative noise, nonlinear invariant measures, nonlinear observables, finite-time nonstationary transitions, codimension-2 bifurcations, and strongly non-normal transient dynamics — and clarifies under which conditions nontrivial observation optimization may survive. The central contribution is a mathematically rigorous no-go/equivalence result that limits the scope of information-geometric optimization claims for early warning signal detection and reframes observation optimization primarily as a finite-time engineering problem rather than a universal asymptotic principle. Author: Diplom-Ingenieur Bernd von Mallinckrodt  E-Mail: [email protected] Keywords:critical transitions, early warning signals, Fisher information, observability, PBH test, stochastic dynamical systems, bifurcation theory, critical slowing down, information geometry, non-normal systems, codimension-1 bifurcation, asymptotic equivalence, stochastic differential equations, covariance dynamics, eigenmode monitoring   English Description This work analyzes the asymptotic relationship between Fisher-information-based observation design and classical eigenmode observability for early warning signals near critical transitions in stochastic dynamical systems. For finite-dimensional stochastic systems with additive noise near generic codimension-1 bifurcations, the paper proves a Rank-1 Cancellation Theorem showing that the Fisher information associated with scalar observations asymptotically collapses to the same observable subspace characterized by Popov–Belevitch–Hautus (PBH) eigenmode observability. Consequently, all scalar observations with nonzero projection onto the critical eigenmode become asymptotically equivalent in Fisher information, while blind directions remain uninformative. The manuscript further investigates potential escape routes — including multiplicative noise, nonlinear invariant measures, nonlinear observables, finite-time nonstationary transitions, codimension-2 bifurcations, and strongly non-normal transient dynamics — and clarifies under which conditions nontrivial observation optimization may survive. The central contribution is a mathematically rigorous no-go/equivalence result that limits the scope of information-geometric optimization claims for early warning signal detection and reframes observation optimization primarily as a finite-time engineering problem rather than a universal asymptotic principle. Author: Diplom-Ingenieur Bernd von Mallinckrodt  E-Mail: [email protected] Keywords:critical transitions, early warning signals, Fisher information, observability, PBH test, stochastic dynamical systems, bifurcation theory, critical slowing down, information geometry, non-normal systems, codimension-1 bifurcation, asymptotic equivalence, stochastic differential equations, covariance dynamics, eigenmode monitoring