Asymptotic properties of the spectrum of neutral delay differential equations

TL;DR

This paper investigates the spectral properties and stability boundaries of neutral delay differential equations as the delay becomes large, providing both analytical and numerical insights into their stability behavior.

Contribution

It introduces an approximation for the stability boundary in large delay regimes and compares it with exact analytical results, enhancing understanding of delay effects on stability.

Findings

Approximate stability boundary aligns well with exact boundary for large delays.

Including delayed velocity feedback improves stability boundary approximation for small delays.

Numerical spectrum computations support the analytical stability analysis.

Abstract

Spectral properties and transition to instability in neutral delay differential equations are investigated in the limit of large delay. An approximation of the upper boundary of stability is found and compared to an analytically derived exact stability boundary. The approximate and exact stability borders agree quite well for the large time delay, and the inclusion of a time-delayed velocity feedback improves this agreement for small delays. Theoretical results are complemented by a numerically computed spectrum of the corresponding characteristic equations.