Exact stability sets for a linear difference system with diagonal delay
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 \((0,\pi/2(k+1))\).
scientific article; zbMATH DE number 5757903