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.摘要 本研究基于瞬态增长与伪谱结构，构建了用于非正则动力学系统不稳定性分析的计算框架。研究表明，基于特征值的经典稳定性准则不足以预测有限时间放大现象。 本文引入无量纲不稳定性泛函，将算子增长与系统容量相联系；通过预解范数估计推导了不稳定性的充分条件，并提出振荡扩展方法以捕捉相位依赖型放大效应。 基于归一化系统指标构建了不稳定性泛函的计算代理模型，验证了理论预测与数值行为的一致性。 所提出的框架为放大机制主导渐近稳定的应用系统的不稳定性分析提供了统一方法。