Advances in Difference Equations
Volume 2006 (2006), Article ID 31409, 9 pages
doi:10.1155/ADE/2006/31409
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
Abstract
We consider the stability problem for the difference system xn=Axn−1+Bxn−k, 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 A≃I and A≃−I.