Stability Of A 2-Dimensional Irreducible Linear System Of Delay Differential Equations

Let r > 0. We consider the system x(t) = Ax(t) Bx(t - r) where A and B are real square matrices of dimension 2. We assume that A has a single eigenvalue and give sufficient conditions for the asymptotic stability of the null solution of the system by deriving a pair of one dimensional delay differential equations from the system and comparing the Lyapunov exponents of the corresponding fundamental solutions.