Stability switches in a linear differential equation with two delays

IF 1 Q1 MATHEMATICS
Y. Hata, H. Matsunaga
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引用次数: 0

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.
具有两个时滞的线性微分方程的稳定性切换
本文研究了时滞对双时滞线性微分方程渐近稳定性的影响 \[x'(t)=-ax(t)-bx(t-\tau)-cx(t-2\tau),\quad t\geq 0,\] 在哪里 \(a\), \(b\),和 \(c\) 是实数和 \(\tau\gt 0\). 建立了该方程零解渐近稳定的若干显式条件。作为一个推论,证明了零解在经历稳定切换有限次后最终变得不稳定 \(\tau\) 只有当 \(c-a\lt 0\) 和 \(\sqrt{-8c(c-a)}\lt |b| \lt a+c\). 显式稳定性依赖于变化 \(\tau\) 还描述了。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Opuscula Mathematica
Opuscula Mathematica MATHEMATICS-
CiteScore
1.70
自引率
20.00%
发文量
30
审稿时长
22 weeks
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