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

Abstract: We bring together three key amplification mechanisms in linear dynamical systems: spectral criticality, resonance, and non-normality. We disentangle and quantitatively couple 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 to both Gaussian noise and periodic forcing in the inertial and overdamped regimes reveal a single combining amplification law, 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. Our framework applies broadly -- from seismology to non-Hermitian photonics and ecological networks -- and offers new diagnostic tools to distinguish true critical behavior from transient amplification driven by non-normality.