Stability switches in a linear differential equation with two delays

This paper is devoted to the study of the effect of delays on the asymptotic stability of a linear differential equation with two delays x′(t) = −ax(t) − bx(t − τ ) − cx(t − 2τ ), t ≥ 0, where a, b, and c are real numbers and τ > 0. We establish some explicit conditions for the zero solution of the equation to be asymptotically stable. As a corollary, it is shown that the zero solution becomes unstable eventually after undergoing stability switches finite times when τ increases only if c−a < 0 and $sqrt{-8c(c-a)} < |b| < a+c$. The explicit stability dependence on the changing τ is also described.