Regimes of Stability in Non-Normal Systems: A Phase Classification Based on the K-Functional

Abstract We develop a phase classification framework for non-normal dynamical systems based on the K-functional and its time evolution. While classical stability theory distinguishes only between stable and unstable regimes, real systems exhibit intermediate behavior characterized by sensitivity, delayed response, and transient amplification. We define three regimes—subcritical, critical, and supercritical—based on the magnitude of the K-functional and its growth rate. We show that instability is preceded by a critical regime in which small perturbations lead to large responses without immediate divergence. The framework provides a refined classification of system behavior and connects operator geometry with observable dynamics.