Table of Contents
ISRN Applied Mathematics
Volume 2011, Article ID 910936, 19 pages
http://dx.doi.org/10.5402/2011/910936
Research Article

The Stability Cone for a Difference Matrix Equation with Two Delays

1Department of Mathematical Analysis, Chelyabinsk State Pedagogical University, Chelyabinsk 454080, Russia
2Applied Mathematics Department, South Ural State University, Chelyabinsk 454080, Russia
3Faculty of Applied Mathematics and Mechanics, Perm State Technical University, Perm 614990, Russia

Received 30 March 2011; Accepted 25 April 2011

Academic Editors: E. Karpov, G. Mishuris, and Y. Qin

Copyright © 2011 S. A. Ivanov et al. 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.

Linked References

  1. Yu. Nikolaev, “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, vol. 65, no. 12, pp. 1904–1914, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. F. M. Dannan, “The asymptotic stability of x(n+k)+ax(n)+bx(nl)=0,” Journal of Difference Equations and Applications, vol. 10, no. 6, pp. 589–599, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. M. Kipnis and R. M. Nigmatulin, “Stability of trinomial linear difference equations with two delays,” Automation and Remote Control, vol. 65, no. 11, pp. 1710–1723, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. S. S. Cheng and S. Y. Huang, “Alternate derivations of the stability region of a difference equation with two delays,” Applied Mathematics E-Notes, vol. 9, pp. 225–253, 2009. View at Google Scholar · View at Zentralblatt MATH
  5. E. Kaslik and St. Balint, “Bifurcation analysis for a two-dimensional delayed discrete-time Hopfield neural network,” Chaos, Solitons & Fractals, vol. 34, no. 4, pp. 1245–1253, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. Matsunaga, “Stability regions for a class of delay difference systems,” in Differences and Differential Equations, vol. 42 of Fields Institute Communications, pp. 273–283, American Mathematical Society, Providence, RI, USA, 2004. View at Google Scholar · View at Zentralblatt MATH
  7. I. S. Levitskaya, “A note on the stability oval for xn+1=xn+Axnk,” Journal of Difference Equations and Applications, vol. 11, no. 8, pp. 701–705, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. E. Kaslik, “Stability results for a class of difference systems with delay,” Advances in Difference Equations, vol. 2009, Article ID 938492, 13 pages, 2009. View at Google Scholar · View at Zentralblatt MATH
  9. J. Diblik and D. Ya. Khusainov, “Representation of solutions of discrete delayed system x(k+1)=Ax(k)+Bx(km)+f(k) with commutative matrices,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 63–76, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. E. N. Gryazina and B. T. Polyak, “Stability regions in the parameter space: D-decomposition revisited,” Automatica, vol. 42, no. 1, pp. 13–26, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. D. Šiljak, “Parameter space methods for robust control design: a guided tour,” IEEE Transactions on Automatic Control, vol. 34, no. 7, pp. 674–688, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. Horn and C. Johnson, Matrix Theory, Cambridge University Press, Cambridge, UK, 1986.
  13. M. Kipnis and V. Malygina, “The stability cone for a matrix delay difference equation,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 860326, 15 pages, 2011. View at Google Scholar
  14. A. Cohn, “Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise,” Mathematische Zeitschrift, vol. 14, no. 1, pp. 110–148, 1922. View at Publisher · View at Google Scholar · View at MathSciNet
  15. S. A. Campbell, Y. Yuan, and S. Bungay, “Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling,” Nonlinearity, vol. 18, no. 6, pp. 2827–2847, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  16. Y. Yuan and S. A. Campbell, “Stability and synchronization of a ring of identical cells with delayed coupling,” Journal of Dynamics and Differential Equations, vol. 16, no. 3, pp. 709–744, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P. J. Davis, Circulant Matrices, AMS Chelsea, New York, NY, USA, 1994.
  18. J. Kaplan and J. Yorke, “Chaotic behavior of multidimensional difference equations,” in Functional Differential Equations and Approximation of Fixed Points, H. O. Peitgen and H. O. Walther, Eds., vol. 730 of Lecture Notes in Mathematics, pp. 204–227, Springer, Berlin, Germany, 2006. View at Google Scholar
  19. M. M. Kipnis and D. A. Komissarova, “Stability of delay difference system,” Advances in Difference Equations, vol. 2006, Article ID 314109, 9 pages, 2006. View at Google Scholar
  20. A. Yu. Kulikov, “Stability of a linear nonautonomous difference equation with bounded delays,” Russian Mathematics, vol. 54, no. 11, pp. 18–26, 2010. View at Publisher · View at Google Scholar
  21. A. Yu. Kulikov and V. V. Malygina, “On the stability of nonautonomous difference equations with several delays,” Russian Mathematics, vol. 52, no. 3, pp. 15–23, 2008. View at Publisher · View at Google Scholar
  22. V. V. Malygina and A. Yu. Kulikov, “On the stability of semi-autonomous difference equations,” Russian Mathematics, vol. 55, no. 5, pp. 19–27, 2011. View at Google Scholar
  23. T. N. Khokhlova, M. M. Kipnis, and V. V. Malygina, “The stability cone for a delay differential matrix equation,” Applied Mathematics Letters, vol. 24, no. 5, pp. 742–745, 2011. View at Publisher · View at Google Scholar
  24. T. L. Sabatulina and V. V. Malygina, “Several stability tests for linear autonomous differential equations with distributed delay,” Russian Mathematics, vol. 51, no. 6, pp. 52–60, 2007. View at Publisher · View at Google Scholar
  25. M. M. Kipnis and I. S. Levitskaya, “Stability of delay difference and differential equations: similarities and distinctions,” in Difference Equations, Special functions and Orthogonal Polynomials, pp. 315–324, World Scientific, Hackensack, NJ, USA, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. I. S. Levitskaya, “Stability domain of a linear differential equation with two delays,” Computers & Mathematics with Applications, vol. 51, no. 1, pp. 153–159, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet