Unifying Framework for Amplification Mechanisms: Criticality, Resonance and Non-Normality

TL;DR

This paper introduces a unified linear framework linking spectral criticality, resonance, and non-normality, providing quantitative tools to analyze and predict amplification effects across various physical and social systems.

Contribution

It presents a novel two-parameter model that unifies amplification mechanisms and offers explicit formulas for system response, expanding understanding of transient amplification phenomena.

Findings

Reveals a universal amplification law with non-normal gain proportional to K^2

Expands conditions for amplification, highlighting a broad pseudo-critical regime

Demonstrates non-normal amplification effects in quantum optics and earthquake models

Abstract

We bring together three key amplification mechanisms in linear dynamical systems: spectral criticality, resonance, and non-normality. We present a unified linear framework that both distinguishes and quantitatively links these effects through two fundamental parameters: (i) the spectral distance to a conventional bifurcation or to a resonance and (ii) a non-normal index $K$ (or condition number $\kappa$) that measures the obliqueness of the eigenvectors. Closed-form expressions for the system's response in the form of the variance $v_\infty$ of the observable responding to both Gaussian noise and periodic forcing reveal a general amplification law $v_\infty = v_0 \left( 1 + \mathcal{G}(K) \right)$ with non-normal gain $\mathcal{G}(K) \propto K^2$ represented in universal phase diagrams. By reanalyzing a model of remote earthquake triggering based on breaking of Hamiltonian symmetry, we illustrate how our two-parameter framework significantly expands both the range of conditions under which amplification can occur and the magnitude of the resulting response, revealing a broad pseudo-critical regime associated with large $\kappa$ that previous single-parameter approaches overlooked. Similarly, in the Non-Hermitian extensions of quantum optics provided by Forward Four-Wave Mixing (FFWM) experiments, we show the presence of a counterintuitive gain-from-loss effect that directly manifests non-normal amplification in a propagating-wave setting. This predicts the possibility to engineer transient optical energy amplification without the need for true lasing or exact $\mathcal{PT}$-symmetry breaking. Our framework applies to many other physical, natural and social systems and offers new diagnostic tools to distinguish true critical behavior from transient amplification driven by non-normality.