Stability switches in a system of linear differential equations with diagonal delay
Stability switches in a system of linear differential equations with diagonal delay
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DOI：
10.1016/j.amc.2009.02.010
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发表时间：
2009-06
期刊：
影响因子：
--
通讯作者：
H. Matsunaga
中科院分区：
文献类型：
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作者：
H. Matsunaga
本文研究了形式为x‘(T)=-ax(t-τ)-by(T)，y’(T)=-cx(T)-ay(t-τ)的时滞微分系统的稳定性问题，其中a，b，c为实数，τ为正数。给出了系统零解渐近稳定的一些充要条件。特别地，当τ从0单调增加时，系统的零解有限次地从稳定切换到不稳定，当0-bc；时从稳定切换到稳定，当-bc；2a<；0时，从不稳定切换到稳定再到不稳定。作为应用，我们研究了时滞Lotka-Volterra系统正平衡点的局部渐近稳定性。
This paper deals with the stability problem of a delay differential system of the form x′(t)=-ax(t-τ)-by(t), y′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if 0<4a<-bc; and from instability to stability to instability if --bc<2a<0. As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka–Volterra systems.