Unified Instability Functional for Non-Normal Dynamical Systems

Abstract We introduce a unified instability functional for non-normal dynamical systems that links transient amplification, system capacity, and perturbation magnitude. Classical eigenvalue-based stability criteria fail to capture transient growth effects inherent in non-normal operators. We define the Kaupp number K = \frac{\|X_0\| G_{\max}}{S}, \quad G_{\max} = \sup_{t \ge 0} \|e^{At}\|, and show that instability occurs when K \ge 1. Using pseudospectral theory and eigenvector conditioning, we establish G_{\max} \sim \sqrt{\kappa(P)}, leading to a geometric formulation of instability. The framework provides a universal criterion for structural failure across dynamical systems.