Analytical Study of Stability and Hopf Bifurcation in Linear Delay Differential Equations

Abstract

This paper investigates stability and delay-induced stability switching phenomena in linear homogeneous delay differential systems. Time delays introduce memory effects that significantly enrich the qualitative behavior of dynamical systems and may destabilize equilibria by generating oscillations. We first establish delay-independent stability criteria based on matrix measures and norm estimates, providing explicit conditions that guarantee uniform exponential stability for all delay values. Next, a spectral analysis of the associated characteristic equation is performed to identify critical delay values at which purely imaginary eigenvalues arise and Hopf-type stability switching occurs. For the two-dimensional case, closed-form analytical expressions for the critical frequency and the sequence of delay thresholds are derived, allowing an exact characterization of stability boundaries. Numerical simulations illustrate the theoretical results and confirm the predicted transitions from stable behavior to sustained oscillations. The proposed framework offers a simple, general, and computationally efficient methodology for analyzing stability and bifurcation mechanisms in linear delay systems and can be naturally extended to weakly nonlinear models and practical applications.

Received: 13 December 2026 / Revised: 21 January 2026 / Accepted: 5 March 2026 / Published: 25 March 2026