Stability of linear delay differential systems via Laplace transformation

We present a Laplace transform approach for explicitly solving linear delay differential systems with multiple discrete delays by applying the Cauchy residue theorem. This method enables direct determination of the stability of the trivial solution when delays are relatively small. Its efficacy is illustrated through two nonlinear models with two and three delays, respectively, for which explicit solutions and stability criteria are obtained. The approach offers two key advantages: (i) analytic solutions are obtained with less effort than the method of steps, and (ii) a small hyper-tetrahedron region in the delay parameter space can be identified in which the trivial solution is asymptotically stable. Furthermore, the results can be combined with existing theory, such as Lemma 3.10 in [1], to establish conditions for Hopf bifurcation. (Dedicated to Professor Shigui Ruan on the occasion of his 60th birthday).