We bring together three key amplification mechanisms in linear dynamical systems: spectral criticality, resonance, and nonnormality. We present a unified linear framework that both distinguishes and quantitatively links these effects through two fundamental parameters: (1) the spectral distance to a conventional bifurcation or to a resonance and (2) a nonnormal index K (or condition number κ ) that measures the obliqueness of the eigenvectors. Closed-form expressions for the system's response in the form of the variance v ∞ of the observable responding to both Gaussian noise and periodic forcing reveal a general amplification law v ∞ = v 0 ( 1 + G ( K ) ) with nonnormal gain G ( K ) ∝ 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 pseudocritical regime associated with large κ that previous single-parameter approaches overlooked. Similarly, in the non-Hermitian extensions of quantum optics provided by forward four-wave mixing experiments, we show the presence of a counterintuitive gain-from-loss effect that directly manifests nonnormal amplification in a propagating-wave setting. This predicts the possibility to engineer transient optical energy amplification without the need for true lasing or exact PT -symmetry breaking. Our framework applies to many other physical, natural, and social systems and offers diagnostic tools to distinguish true critical behavior from transient amplification driven by nonnormality.
Troude et al. (Wed,) studied this question.