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Zbl 1067.39012
Matsunaga, Hideaki
Stability regions for a class of delay difference systems.
(English)
[A] Elaydi, Saber (ed.) et al., Difference and differential equations. Proceedings of the 7th international conference on difference equations and applications (ICDEA), Changsha, China, August 12--17, 2002. Providence, RI: American Mathematical Society (AMS). Fields Institute Communications 42, 273-283 (2004). ISBN 0-8218-3354-5/hbk

Summary: We give some new necessary and sufficient conditions for the asymptotic stability of a linear delay difference system $$x_{n+1}- ax_n+ Bx_{n-k}= 0,\quad n= 0,1,2,\dots,\tag L$$ by using root-analysis or its characteristic equation. Here, $a$ is a real number, $B$ is a $2\times 2$ real constant matrix, and $k$ is a positive integer. In case $$B= b\pmatrix \cos\theta &-\sin\theta\\ \sin\theta & \cos\theta\endpmatrix,\quad b\in\bbfR,\ |\theta|\le{\pi\over 2},$$ the stability region for (L), that is, the region of $(a, b)$ for which the zero solution of (L) with fixed $k$ and $\theta$ is asymptotically stable, is symmetric with respect to the $b$-axis (resp. origin) if $k$ is odd (resp. even).
MSC 2000:
*39A11 Stability of difference equations

Keywords: asymptotic stability; linear delay difference system

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