Unifying Dynamical and Structural Stability of Equilibriums

TL;DR

This paper establishes a fundamental link between dynamical and structural stability of equilibriums in real dynamical systems, showing that a system's response to stochastic noise reflects its internal stability limits.

Contribution

It demonstrates that the local response to white-noise perturbations accurately indicates the system's internal stability threshold, unifying dynamical and structural stability concepts.

Findings

Dynamical stability equals the minimal internal perturbation destabilizing the equilibrium.

Harmonic external perturbations relate to spectral sensitivity of the Jacobian.

White-noise response reflects the system's internal noise tolerance.

Abstract

We exhibit a fundamental relationship between measures of dynamical and structural stability of equilibriums, arising from real dynamical systems. We show that dynamical stability, quantified via systems local response to external perturbations, coincides with the minimal internal perturbation able to destabilize the equilibrium. First, by reformulating a result of control theory, we explain that harmonic external perturbations reflect the spectral sensitivity of the Jacobian matrix at the equilibrium, with respect to constant changes of its coefficients. However, for this equivalence to hold, imaginary changes of the Jacobian's coefficients have to be allowed. The connection with dynamical stability is thus lost for real dynamical systems. We show that this issue can be avoided, thus recovering the fundamental link between dynamical and structural stability, by considering stochastic noise as external and internal perturbations. More precisely, we demonstrate that a system's local response to white-noise perturbations directly reflects the intensity of internal white noise that it can accommodate before asymptotic mean-square stability of the equilibrium is lost.