Reply to Comment on “Synchronization dynamics in non-normal networks: the trade-off for optimality”

The paper’s reply letter argues qualitatively that in nonlinear systems with a non-normal Hurwitz Jacobian, small but finite perturbations can experience transient growth large enough to exit the basin of attraction, so linear (local) stability can fail to predict the eventual outcome; it ties this to pseudospectral bounds and the Kreiss constant K and explicitly emphasizes “small but finite” perturbations, not arbitrarily small ones (see their discussion of supt ||e^{tJ}|| and K, and of basin shrinkage) . The candidate model solution rejects a stronger, non-claimed statement (“for every sufficiently small ε…”), explains why it would contradict local exponential stability, and then gives a rigorous corrected threshold result quantifying finite-perturbation fragility using variation of constants, a local Lipschitz bound for DF, the decay integral B = ∫0^∞ ||e^{sJ}|| ds, and the Kreiss constant K. This aligns with the paper’s qualitative position and literature quotes that stress small-but-finite perturbations and basin reduction, e.g., their citations and quotes in the reply letter . Net: the paper’s qualitative claim is correct (though non-rigorous), and the model’s solution supplies a compatible, quantitative proof framework; there is no direct contradiction.