Pseudospectral Precursors of Systemic Failure: Early Warning Signals in Non-Normal Systems

Abstract Classical stability analysis relies on spectral properties and asymptotic behavior, which do not capture finite-time amplification in non-normal systems. As a result, systems deemed stable may exhibit significant transient growth prior to any asymptotic decay. This work develops a pseudospectral framework for identifying early warning signals of instability. It is shown that the geometry of the ε-pseudospectrum provides a leading indicator of transient amplification and impending capacity exceedance. A connection is established between resolvent growth, transient amplification, and a dimensionless instability functional. A criterion is derived linking the extension of the pseudospectrum into the right half-plane with the onset of finite-time instability. Numerical examples demonstrate that pseudospectral expansion precedes the crossing of the critical instability threshold, providing a predictive signal unavailable in classical spectral analysis.