Asymptotic constancy for systems of delay differential equations. (English) Zbl 0959.34058

Murakami, Kouichi

Nonlinear Anal., Theory Methods Appl. 30, No. 7, 4595-4606 (1997).

Results of the paper of K. L. Cooke and J. A. Yorke [Math. Biosci. 16, 75-101 (1973; Zbl 0251.92011)] and the author’s papers [Funkc. Ekvacioj., Ser. Int. 39, No. 3, 519-540 (1996; Zbl 0872.34052) and J. Math. Anal. Appl. 205, No. 2, 512-530 (1997; Zbl 0885.34060)] are extended to the \(n\)-dimensional system \[ x'(t)= A\bigl(x(t- \tau_1)-x(t-\tau _2)\bigr), \quad A\in \mathbb{R}^{n\times n},\;0<\tau_1<\tau_2. \] For an arbitrary \(2\times 2\)-matrix the author gives precise tests for the asymptotic behavior of the solutions depending on \(A\). The solutions may approach a constant or may approach a periodic orbit, or may be unbounded. For the \(n\)-dimensional systems conditions for the existence of an asymptotic equilibrium point are given.

Reviewer: Pyotr M.Simonov (MR 99f:34106)

MSC:

34K13	Periodic solutions to functional-differential equations
34K11	Oscillation theory of functional-differential equations
34K20	Stability theory of functional-differential equations

Keywords:

systems of delay differential equations; solutions; periodic orbit; existence; asymptotic equilibrium

Citations:

Zbl 0251.92011; Zbl 0872.34052; Zbl 0885.34060

References:

[1]	Cooke, K.; Yorke, J., Some Equations Modelling Growth Processes and Gonorrhea Epidemics, Math. Biosci., 16, 75-101 (1973) · Zbl 0251.92011
[2]	Diekmann, O.; Van Gils, S. A.; Lunel, S. M. Verduyn; Walther, H.-O., (Delay Equations: Functional-, Complex-, and Nonlinear Analysis (1995), Springer-Verlag: Springer-Verlag Warszawa) · Zbl 0826.34002
[3]	Hale, J., (Theory of Functional Differential Equations (1977), Springer-Verlag)
[4]	Hale, J.; Lunel, S. M. Verduyn, (Introduction to Functional Differential Equations (1993), Springer-Verlag) · Zbl 0787.34002
[5]	Kuang, Y., (Delay Differential Equations With Applications in Population Dynamics (1993), Academic Press)
[6]	Murakami, K., Asymptotic Constancy and Periodic Solutions for Linear Autonomous Delay Differential Equation, Funkcial. Ekvac., 39, 519-540 (1996) · Zbl 0872.34052