Stability Transitions in Meta-Hierarchical Systems: A Non-Normal Dynamics Perspective

Abstract Stability in large-scale coupled systems is traditionally assessed through spectral analysis of linearized operators. However, it is now well-established that spectral stability does not guarantee bounded transient behavior in non-normal systems. In this work, we introduce a universal stability criterion based on the balance between transient amplification, system perturbations, and structural capacity. We define the Kaupp Stability Number K = \frac{M \cdot \|\Delta X\|}{R} where M = \sup_{t \ge 0} \|e^{At}\| is the transient amplification factor, \|\Delta X\| represents aggregate perturbations, and R denotes the operational capacity of the system. We prove that the condition K \le 1 is a necessary constraint for structural stability in non-normal systems. The proposed framework applies universally across engineered, physical, and coupled networked systems, providing a geometric interpretation of instability as a consequence of operator non-normality rather than spectral divergence.