Exact stability sets for a linear difference system with diagonal delay
type: Item
instance of: scholarly article
title: Exact stability sets for a linear difference system with diagonal delay
zbMATH Open document ID: 1198.39027
author: H. Matsunaga
author: Chikako Hajiri
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-07-20T00:00:00Z
review text: The article deals with the following delay difference system \[ x_{n+1} = ax_{n-k} + by_n, \quad y_{n+1} = cx_n + ay_{n-k}, \qquad n = 0,1,2,\dots, \] where \(a, b, c\) are reals and \(k\) is a positive integer. The main results are two theorems about the asymptotic stability of the zero solutions of this system. More precisely, the authors prove: if \(k\) is even, then the zero solution is asymptotically stable if and only if \(\sqrt{|bc|} < (k + 1)/k\) and \[ \begin{cases} -(bc + 1 - 2\sqrt{bc} \cos \phi)^\frac12 < a < 1 - \sqrt{bc} &\text{for } bc \geq 0,\\ -(- bc + 1 - 2\sqrt{-bc} \cos \phi)^\frac12 < (-1)^{(k+1)/2}a < 1 - \sqrt{-bc} &\text{for } bc < 0,\end{cases} \] where \(\phi\) is the solution of the equation \(\sqrt{|bc|} \sin k\phi = \sin (k + 1)\phi\) in \((0,\pi/(k+1))\); similarly, if \(k\) is odd, then the zero solution is asymptotically stable if and only if \(\sqrt{|bc|} < 1\) and \[ \begin{cases} |a| < 1 \sqrt{bc} &\text{for } bc \geq 0,\\ |a| < (- bc + 1 - 2\sqrt{-bc} \cos \psi)^\frac12 &\text{for } bc < 0,\end{cases} \] where \(\psi\) is the solution of the equation \(\sqrt{|bc|} \cos k\psi = \cos (k + 1)\psi\) in \((0,\pi/2(k+1))\).
Mathematics Subject Classification ID: 39A30
Mathematics Subject Classification ID: 39A06
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34K06
zbMATH DE Number: 5757903
zbMATH Keywords: delay difference systems
zbMATH Keywords: discrete systems
zbMATH Keywords: stability
zbMATH Keywords: systems with diagonal delay
zbMATH Keywords: asymptotic stability
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.03.062
OpenAlex ID: W2064016788
cites work: Q4679023
cites work: Stability and bifurcation in a discrete system of two neurons with delays
cites work: Bifurcation analysis for a two-dimensional delayed discrete-time hopfield neural network
cites work: Chaotic dynamics of a delayed discrete-time Hopfield network of two nonidentical neurons with no self-connections
cites work: A note on explicit stability conditions for autonomous higher order difference equations
cites work: Stability of the trinomial linear difference equations with two delays
cites work: A note on unbounded solutions of a class of second order rational difference equations
cites work: The asymptotic stability of \(x_{n+1}-ax_ n+bx_{n-k}=0\)
cites work: A note on the stability oval for
cites work: Q4676210
cites work: Q4944361
cites work: Asymptotic Behaviors of a Linear Difference System with Multiple Delays
cites work: Stability Analysis of Second Order Delay Difference Equations
cites work: Stability and bifurcation of a two-dimension discrete neural network model with multi-delays
DOI: 10.1016/J.JMAA.2010.03.062
description: scientific article; zbMATH DE number 5757903