Unified Theory of Capacity-Constrained Instability Non-Normal Dynamics, Pseudospectra, and Systemic Risk

Abstract This paper develops a computational framework for instability analysis in non-normal dynamical systems based on transient growth and pseudospectral structure. Classical eigenvalue-based stability criteria are shown to be insufficient for predicting finite-time amplification. A dimensionless instability functional is introduced, linking operator growth to system capacity. Sufficient conditions for instability are established using resolvent norm estimates. An oscillatory extension is proposed to capture phase-dependent amplification effects. A computational proxy for the instability functional is constructed using normalized system indicators, demonstrating consistency between theoretical predictions and numerical behavior. The proposed framework provides a unified approach for analyzing instability in applied systems where amplification mechanisms dominate over asymptotic stability.