Open Access

Stability of a delay difference system

Advances in Difference Equations20062006:031409

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

Received: 28 January 2006

Accepted: 1 June 2006

Published: 2 August 2006

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.

[12345678910111213141516]

Authors’ Affiliations

(1)
Department of Mathematics, Chelyabinsk State Pedagogical University
(2)
Department of Mathematics, Southern Ural State University

References

  1. Berezansky L, Braverman E: On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations. Journal of Mathematical Analysis and Applications 2005,304(2):511–530. 10.1016/j.jmaa.2004.09.042MathSciNetView ArticleMATHGoogle Scholar
  2. Berezansky L, Braverman E, Liz E: Sufficient conditions for the global stability of nonautonomous higher order difference equations. Journal of Difference Equations and Applications 2005,11(9):785–798. 10.1080/10236190500141050MathSciNetView ArticleMATHGoogle Scholar
  3. Cohn A: Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Mathematische Zeitschrift 1922,14(1):110–148. 10.1007/BF01215894MathSciNetView ArticleMATHGoogle Scholar
  4. Cooke KL, Györi I: Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments. Computers & Mathematics with Applications 1994,28(1–3):81–92.View ArticleMathSciNetMATHGoogle Scholar
  5. Dannan FM: The asymptotic stability of x ( n + k )+ ax ( n )+ bx ( n - l ) = 0. Journal of Difference Equations and Applications 2004,10(6):589–599. 10.1080/10236190410001685058MathSciNetView ArticleMATHGoogle Scholar
  6. Diblík J, Khusainov DYa: Representation of solutions of discrete delayed system x ( k +1) = Ax ( k ) + Bx ( k - m ) + f ( k ) with commutative matrices. Journal of Mathematical Analysis and Applications 2006,318(1):63–76. 10.1016/j.jmaa.2005.05.021MathSciNetView ArticleMATHGoogle Scholar
  7. Kipnis M, Nigmatulin RM: Stability of trinomial linear difference equations with two delays. Automation and Remote Control 2004,65(11):1710–1723.MathSciNetView ArticleMATHGoogle Scholar
  8. Kuruklis SA: The asymptotic stability of xn+1- ax n + bxn-k = 0. Journal of Mathematical Analysis and Applications 1994,188(3):719–731. 10.1006/jmaa.1994.1457MathSciNetView ArticleMATHGoogle Scholar
  9. Levin SA, May RM: A note on difference-delay equations. Theoretical Population Biology 1976,9(2):178–187. 10.1016/0040-5809(76)90043-5MathSciNetView ArticleMATHGoogle Scholar
  10. Levitskaya IS: A note on the stability oval for xn+ = x n + Axn-k. Journal of Difference Equations and Applications 2005,11(8):701–705. 10.1080/10236190512331333851MathSciNetView ArticleMATHGoogle Scholar
  11. Liz E, Ferreiro JB: A note on the global stability of generalized difference equations. Applied Mathematics Letters 2002,15(6):655–659. 10.1016/S0893-9659(02)00024-1MathSciNetView ArticleMATHGoogle Scholar
  12. Liz E, Pituk M: Asymptotic estimates and exponential stability for higher-order monotone difference equations. Advances in Difference Equations 2005,2005(1):41–55. 10.1155/ADE.2005.41MathSciNetView ArticleMATHGoogle Scholar
  13. Nikolaev YuP: The set of stable polynomials of linear discrete systems: its geometry. Automation and Remote Control 2002,63(7):1080–1088. 10.1023/A:1016154714222MathSciNetView ArticleMATHGoogle Scholar
  14. Nikolaev YuP: The geometry of D -decomposition of a two-dimensional plane of arbitrary coefficients of the characteristic polynomial of a discrete system. Automation and Remote Control 2004,65(12):1904–1914.MathSciNetView ArticleMATHGoogle Scholar
  15. Papanicolaou VG: On the asymptotic stability of a class of linear difference equations. Mathematics Magazine 1996,69(1):34–43. 10.2307/2691392MathSciNetView ArticleMATHGoogle Scholar
  16. Rekhlitskii ZI: On the stability of solutions of certain linear differential equations with a lagging argument in the Banach space. Doklady Akademii Nauk SSSR 1956, 111: 770–773.Google Scholar

Copyright

© M. Kipnis and D. Komissarova. 2006

This article is published under license to BioMed Central Ltd. 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.