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Stability switches in a linear differential equation with two delays

Hata Yuki, Matsunaga Hideaki

Opuscula Mathematica

2022

Vol. 42, no. 5

673--690

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 [formula]. The explicit stability dependence on the changing τ is also described.