Stability switches in a linear differential equation with two delays

Abstract

This paper is devoted to the study of the effect of delays on the asymptotic stability of a linear differential equation with two delays $x'(t)=-ax(t)-bx(t-\tau)-cx(t-2\tau),\quad t\geq 0,$ where $a$, $b$, and $c$ are real numbers and $\tau \gt 0$. 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 $\tau$ increases only if $c-a\lt 0$ and $\sqrt{-8c(c-a)}\lt |b| \lt a+c$. The explicit stability dependence on the changing $\tau$ is also described.