Dynamical Evolution of the K-Functional: A Differential Framework for Instability Onset

Abstract We introduce a dynamical framework for the evolution of instability in non-normal systems based on the time-dependent K-functional. While existing approaches focus on static stability criteria, the present work defines a differential measure governing the rate at which a system approaches instability. We derive the evolution equation \mathcal{T}(t) = \frac{d}{dt}\ln K(t), and show that instability is characterized not only by the magnitude of K, but also by its growth rate. The framework provides a predictive criterion for instability onset and connects transient amplification, perturbation growth, and structural changes within a unified dynamical model.