Opuscula Math. 42, no. 5 (2022), 673-690
https://doi.org/10.7494/OpMath.2022.42.5.673
Opuscula Mathematica
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.
Keywords: delay differential equations, stability switches, two delays.
Mathematics Subject Classification: 34K06, 34K20, 34K25.
- D.M. Bortz, Characteristic roots for two-lag linear delay differential equations, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), 2409-2422. https://doi.org/10.3934/dcdsb.2016053
- B. Cahlon, D. Schmidt, Stability criteria for first order delay differential equations with M commensurate delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17 (2010), 157-177.
- K.L. Cooke, Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 (1982), 592-627. https://doi.org/10.1016/0022-247X(82)90243-8
- T. Elsken, The region of (in)stability of a 2-delay equation is connected, J. Math. Anal. Appl. 261 (2001), 497-526. https://doi.org/10.1006/jmaa.2001.7536
- K. Gu, S. Niculescu, J. Chen, On stability crossing curves for general systems with two delays, J. Math. Anal. Appl. 311 (2005), 231-253. https://doi.org/10.1016/j.jmaa.2005.02.034
- J.K. Hale, W. Huang, Global geometry of the stable regions for two delay differential equations, J. Math. Anal. Appl. 178 (1993), 344-362. https://doi.org/10.1006/jmaa.1993.1312
- J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-4342-7
- N.D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Soc. 25 (1950), 226-232. https://doi.org/10.1112/jlms/s1-25.3.226
- Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
- J.M. Mahaffy, T.C. Busken, Regions of stability for a linear differential equation with two rationally dependent delays, Discrete Contin. Dyn. Syst. 35 (2015), 4955-4986. https://doi.org/10.3934/dcds.2015.35.4955
- J.M. Mahaffy, K.M. Joiner, P.J. Zak, A geometric analysis of stability regions for a linear differential equation with two delays, Internat. J. Bifur. Chaos 5 (1995), 779-796. https://doi.org/10.1142/S0218127495000570
- H. Matsunaga, Delay-dependent and delay-independent stability criteria for a delay differential system, Proc. Amer. Math. Soc. 136 (2008), 4305-4312. https://doi.org/10.1090/S0002-9939-08-09396-9
- H. Matsunaga, Stability switches in a system of linear differential equations with diagonal delay, Appl. Math. Comput. 212 (2009), 145-152. https://doi.org/10.1016/j.amc.2009.02.010
- J. Nishiguchi, On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay, Discrete Contin. Dyn. Syst. 36 (2016), 5657-5679. https://doi.org/10.3934/dcds.2016048
- S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (2003), 863-874.
- G.M. Schoen, H.P. Geering, Stability condition for a delay differential system, Internat. J. Control 58 (1993), 247-252. https://doi.org/10.1080/00207179308923000
- G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific & Technical, New York, 1989.
- X. Yan, J. Shi, Stability switches in a logistic population model with mixed instantaneous and delayed density dependence, J. Dynam. Differential Equations 29 (2017), 113-130. https://doi.org/10.1007/s10884-015-9432-3
- X. Zhang, J. Wu, Critical diapause portion for oscillations: parametric trigonometric functions and their applications for Hopf bifurcation analyses, Math. Methods Appl. Sci. 42 (2019), 1363-1376. https://doi.org/10.1002/mma.5424
- 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
- Communicated by Josef Diblík.
- Received: 2022-04-25.
- Revised: 2022-05-27.
- Accepted: 2022-06-14.
- Published online: 2022-09-08.
