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Stability of a delay difference system

Advances in Difference Equations volume 2006, Article number: 031409 (2006) Cite this article

Abstract

We consider the stability problem for the difference system x n = 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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Authors and Affiliations

  1. Department of Mathematics, Chelyabinsk State Pedagogical University, 69 Lenin Avenue, Chelyabinsk, 454080, Russia

    Mikhail Kipnis

  2. Department of Mathematics, Southern Ural State University, 76 Lenin Avenue, Chelyabinsk, 454080, Russia

    Darya Komissarova

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  1. Mikhail Kipnis
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  2. Darya Komissarova
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Correspondence to Mikhail Kipnis.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kipnis, M., Komissarova, D. Stability of a delay difference system. Adv Differ Equ 2006, 031409 (2006). https://doi.org/10.1155/ADE/2006/31409

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  • DOI: https://doi.org/10.1155/ADE/2006/31409

Keywords

  • Differential Equation
  • Positive Integer
  • Partial Differential Equation
  • Stability Condition
  • Ordinary Differential Equation