Opuscula Mathematica
Opuscula Math. 42, no. 5 (2022), 673-690

Stability switches in a linear differential equation with two delays

Yuki Hata
Hideaki Matsunaga
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.
Keywords: delay differential equations, stability switches, two delays.
Mathematics Subject Classification: 34K06, 34K20, 34K25.
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About the authors ▼
Yuki Hata
Osaka Prefecture University, Department of Mathematical Sciences, Sakai 599-8531, Japan
Hideaki Matsunaga (corresponding author)
https://orcid.org/0000-0001-5805-2303
Osaka Metropolitan University, Department of Mathematics, Sakai 599-8531, Japan
About the article ▼
Communicated by Josef Diblík.
Received: 2022-04-25.
Revised: 2022-05-27.
Accepted: 2022-06-14.
Published online: 2022-09-08.
Cite this article as ▼
Yuki Hata, Hideaki Matsunaga, Stability switches in a linear differential equation with two delays, Opuscula Math. 42, no. 5 (2022), 673-690, https://doi.org/10.7494/OpMath.2022.42.5.673.
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