Exact stability sets for a linear difference system with diagonal delay

Abstract

This paper is devoted to the stability analysis of a delay difference system of the form x n+1 = ax n−k + by n , y n+1 = cx n + ay n−k , where a, b and c are real numbers and k is a positive integer. We establish some exact conditions for the zero solution of the system to be asymptotically stable.

Two types of stability conditions for linear delay difference equations

“…In addition to Kuruklis' pioneering paper [13], where this type of conditions appeared for the first time, and papers [6], [18], which have been already mentioned, we can refer to papers by Dannan [5] and Matsunaga and Hajiri [16]. In addition to Kuruklis' pioneering paper [13], where this type of conditions appeared for the first time, and papers [6], [18], which have been already mentioned, we can refer to papers by Dannan [5] and Matsunaga and Hajiri [16].…”

“…Therefore, it might be useful to reformulate the existing results of papers [5], [16] into such explicit conditions and, of course, to discuss these conditions for other forms of delay difference equations, which have not been considered yet. Therefore, it might be useful to reformulate the existing results of papers [5], [16] into such explicit conditions and, of course, to discuss these conditions for other forms of delay difference equations, which have not been considered yet.…”

“…To summarize, Theorem 3.6 (supplied with Remark 3.8) and Remark 3.9 imply that (40) is asymptotically stable if and only if k ∈ {3, 4,5,7,8,9,11,12,13,16,20,24, 28} . Further, if both k and k/2 are even, then k < 29.6158 due to Remark 3.9.…”

On a three term linear difference equation with complex coefficients

“…Now we apply Theorem 2.3 to (13). It occurs the case (ii) of Theorem 2.3, where the left-hand side of (5) represents the linear function of variable k and the right-hand side of (5) represents the piecewise linear function of variable k. While the left-hand side of (5) increases, the right-hand side of (5) is equal to π (the case k even), or zero (the case k odd).…”

“…We note that many other relevant results on the asymptotic stability of higher order linear difference equation utilizes the conditions of the type (3), (4), (see [8]- [10], [12], [13]; for other related results we refer to [3]- [7], [11], [14], [15]). Our opinion is that also these conditions can be reformulated via conditions of the type (5).…”