Stability switches in linear delay difference equations. (English) Zbl 1335.39024
Summary: The paper discusses necessary and sufficient conditions for the asymptotic stability of the zero solution of the linear delay difference equation \(y(n + 1) = \alpha y(n) + \beta y(n - k)\), where \(\alpha, \beta\) are complex numbers and \(k\) is a positive integer. Compared to the case when \(\alpha, \beta\) are real numbers, the stability behavior of this equation turns out to be much richer. In particular, if \(| \alpha | + | \beta | > 1\) then, as \(k\) monotonously increases, the equation may switch finite times from asymptotic stability to instability and vice versa. We describe an interesting structure of the set of these stability switches, their explicit values and apply the obtained results to some important delay difference equations and their systems.
MSC:
| 39A30 | Stability theory for difference equations |
References:
| [1] | Berezansky, L.; Braverman, E.; Liz, E., Sufficient conditions for the global stability of nonautonomous higher order difference equations, J. Differ. Equ. Appl., 11, 785-798 (2005) · Zbl 1078.39005 |
| [2] | Braverman, E.; Karpuz, B., On stability of delay difference equations with variable coefficients: successive products tests, Adv. Differ. Equ., 2012 (2012), 8 pp · Zbl 1377.39029 |
| [3] | Čermák, J., Asymptotic bounds for linear difference systems, Adv. Differ. Equ., 2010 (2010), Article ID 182696, 14 pp · Zbl 1191.39016 |
| [4] | Čermák, J.; Jánský, J.; Kundrát, P., On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations, J. Differ. Equ. Appl., 18, 1781-1800 (2012) · Zbl 1261.39022 |
| [5] | Čermák, J.; Tomášek, P., On delay-dependent stability conditions for a three-term linear difference equation, Funkcial. Ekvac., 57, 91-106 (2014) · Zbl 1294.39011 |
| [6] | Cheng, S. S.; Huang, S. Y., Alternate derivations of the stability region of a difference equation with two delays, Appl. Math. E-Notes, 9, 225-253 (2009) · Zbl 1204.39015 |
| [7] | Diblík, J.; Růžičková, M.; Šutá, Z., Asymptotic convergence of the solutions of a discrete equation with several delays, Appl. Math. Comput., 218, 5391-5401 (2012) · Zbl 1302.39011 |
| [8] | Freedman, H. I.; Kuang, Y., Stability switches in linear scalar neutral delay equations, Funkcial. Ekvac., 34, 187-209 (1991) · Zbl 0749.34045 |
| [9] | Győri, I.; Horváth, L., Existence of periodic solutions in a linear higher order system of difference equations, Comput. Math. Appl., 66, 2239-2250 (2013) · Zbl 1350.39007 |
| [10] | Ivanov, S.; Kipnis, M. M.; Malygina, V. V., The stability cone for a difference matrix equation with two delays, ISRN Appl. Math., 2011 (2011), Article ID 910936, 19 pp · Zbl 1242.39027 |
| [11] | Kaslik, E., Stability results for a class of difference systems with delay, Adv. Differ. Equ., 2009 (2009), Article ID 938492, 13 pp · Zbl 1187.39025 |
| [12] | Kipnis, M. M.; Malygina, V. V., The stability cone for a matrix delay difference equation, Int. J. Math. Math. Sci., 2011 (2011), Article ID 860326, 15 pp · Zbl 1216.39020 |
| [13] | Kipnis, M. M.; Nigmatullin, R. M., Stability of the trinomial linear difference equations with two delays, Autom. Remote Control, 65, 11, 1710-1723 (2004) · Zbl 1080.39013 |
| [14] | Kuruklis, S. A., The asymptotic stability of \(x_{n + 1} - ax_n + bx_{n - k} = 0\), J. Math. Anal. Appl., 188, 719-731 (1994) · Zbl 0842.39004 |
| [15] | Liz, E., On explicit conditions for the asymptotic stability of linear higher order difference equations, J. Math. Anal. Appl., 303, 492-498 (2005) · Zbl 1068.39017 |
| [16] | Marden, M., Geometry of Polynomials (1966), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0173.02703 |
| [17] | Matsunaga, H., Stability switches in a system of linear differential equations with diagonal delay, Appl. Math. Comput., 212, 145-152 (2009) · Zbl 1171.34346 |
| [18] | Matsunaga, H.; Hajiri, C., Exact stability sets for a linear difference system with diagonal delay, J. Math. Anal. Appl., 369, 616-622 (2010) · Zbl 1198.39027 |
| [19] | Matsunaga, H.; Hashimoto, H., Asymptotic stability and stability switches in a linear integro-differential system, Differ. Equ. Appl., 3, 43-55 (2011) · Zbl 1221.34200 |
| [20] | Papanicolaou, V. G., On the asymptotic stability of a class of linear difference equations, Math. Mag., 69, 1, 34-43 (1996) · Zbl 0866.39001 |
| [21] | Stević, S., A note on recursive sequence \(x_{n + 1} = p_k x_n + p_{k - 1} x_{n - 1} + \cdots + p_1 x_{n - k + 1}\), Ukr. Math. J., 55, 691-697 (2003) · Zbl 1077.39009 |
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