Torus Lifting and Frequency Collisions in the Strong Stability of Neutral Delay Systems.

Abstract We investigate the strong stability of linear neutral functional differential equations with multiple discrete delays, focusing on the existence of a uniform spectral gap separating the characteristic roots from the imaginary axis. We show that the loss of this spectral gap is governed by a structural dynamical mechanism: distinct coupling patterns in the characteristic determinant may generate identical frequency vectors, leading to cancellations through frequency collisions. Assuming rational independence of the delays, we lift the spectral problem to a multidimensional torus, where it can be interpreted in terms of frequency interactions on a compact dynamical system and analyzed via the combinatorics of the determinant expansion. This leads to the introduction of a graph-theoretic condition, called Pure Structural Injectivity (PSI), which prevents such collisions and depends only on the delay-labeled interaction graph. We prove that PSI is necessary for the existence of a uniform spectral gap, while under PSI and a standard contractivity condition the system admits a robust spectral gap. The framework extends to rationally dependent delays via invariant subtori, and in dimension three we provide a complete classification of structurally stable coupling patterns.