Stability of a delay difference system

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.

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