A note on asymptotic stability of delay difference systems
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For the linear delay difference system xn+1 − xn = Axn−k , where A is a 2 × 2 real constant matrix and k is a nonnegative integer, we present an explicit necessary and sufficient condition for the asymptotic stability of the zero solution of this system in terms of det A, trA, and the delay k. 1. Introduction In this paper, we are concerned with the asymptotic stability of the zero solution of the linear delay difference system xn+1 − xn = Axn−k ,
n = 0,1,2,...,
(1.1)
where A is a 2 × 2 real constant matrix and k is a nonnegative integer. In the scalar case, Levin and May [7] showed that the zero solution of the delay difference equation xn+1 − xn = −axn−k is asymptotically stable if and only if 0 < a < 2sin
π/2 kπ . = 2cos 2k + 1 2k + 1
(1.2)
This nice result is proved by using the fact that the zero solution of the linear difference equation is asymptotically stable if and only if all the roots of its associated characteristic equation are inside the unit disk. Here, the Schur-Cohn criterion (see [2, 5]) and the Jury criterion (see [3]) are known to be effective tools for determining the asymptotic stability of linear difference systems. However, several kinds of the necessary and sufficient conditions established by the above criteria are too much complicated even to verify the condition (1.2). In fact, we need some careful root analysis of the characteristic equation in and on the unit circle to get the condition (1.2); see [6, 7, 8]. The purpose of this paper is to give an explicit necessary and sufficient condition for the asymptotic stability of the zero solution of the system (1.1) in terms of detA, trA, and the delay k. As an application, we investigate the local asymptotic stability of delay difference systems of Lotka-Volterra type. For the general background of delay difference systems, one can refer to recent books [1, 2, 4]. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:2 (2005) 119–125 DOI: 10.1155/JIA.2005.119
120
A note on asymptotic stability of delay difference systems
2. Main result Our main result is stated as follows. Theorem 2.1. The zero solution of (1.1) is asymptotically stable if and only if
−1
√
2 detA sin (2k + 1)sin
detA 2
< − trA < 2sin
0 < detA < 4sin2
π/2 detA , + 2k + 1 2sin (π/2)/(2k + 1)
π/2 . 2k + 1
(2.1)
Remark 2.2. In case A = diag[−a, −a], one can √ easily verify that the condition (2.1) is equivalent to the condition (1.2) because of 2 detA = − trA. Remark 2.3. In case k = 1, it follows from Theorem 2.1 that the zero solution of (1.1) is asymptotically stable if and only if −(detA)2 + 3detA < − trA < 1 + detA,
(2.2)
0 < detA < 1.
Remark 2.4. Let k = 0 and let A = B − I, where B is a 2 × 2 real constant matrix and I is the 2 × 2 indentity matrix. Then one can easily see that the system (1.1) becomes xn+1 = Bxn
(2.3)
and the condition (2.1) is reduced to 1 − detB > 0, 1 + trB + detB > 0,
(2.4)
1 − trB + detB > 0, namely, | tr B | < 1 + detB < 2
(2.5)
because detA = detB − trB + 1 and trA = trB − 2
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