Advances in Difference Equations
Volume 2006 (2006), Article ID 31409, 9 pages

Stability of a delay difference system

Mikhail Kipnis1 and Darya Komissarova2

1Department of Mathematics, Chelyabinsk State Pedagogical University, 69 Lenin Avenue, Chelyabinsk 454080, Russia
2Department of Mathematics, Southern Ural State University, 76 Lenin Avenue, Chelyabinsk 454080, Russia

Received 28 January 2006; Revised 22 May 2006; Accepted 1 June 2006

Copyright © 2006 Mikhail Kipnis and Darya Komissarova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the stability problem for the difference system xn=Axn1+Bxnk, where A, B are real matrixes and the delay k is a positive integer. In the case A=I, the equation is asymptotically stable if and only if all eigenvalues of the matrix B lie inside a special stability oval in the complex plane. If k is odd, then the oval is in the right half-plane, otherwise, in the left half-plane. If A+B<1, then the equation is asymptotically stable. We derive explicit sufficient stability conditions for AI and AI.