Abstract and Applied Analysis

An Algebraic Criterion of Zero Solutions of Some Dynamic Systems

Ying Wang, Baodong Zheng, and Chunrui Zhang

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Abstract

We establish some algebraic results on the zeros of some exponential polynomials and a real coefficient polynomial. Based on the basic theorem, we develop a decomposition technique to investigate the stability of two coupled systems and their discrete versions, that is, to find conditions under which all zeros of the exponential polynomials have negative real parts and the moduli of all roots of a real coefficient polynomial are less than 1.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 956359, 13 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364476010

Digital Object Identifier
doi:10.1155/2012/956359

Mathematical Reviews number (MathSciNet)
MR3004920

Zentralblatt MATH identifier
1260.34140

Citation

Wang, Ying; Zheng, Baodong; Zhang, Chunrui. An Algebraic Criterion of Zero Solutions of Some Dynamic Systems. Abstr. Appl. Anal. 2012 (2012), Article ID 956359, 13 pages. doi:10.1155/2012/956359. https://projecteuclid.org/euclid.aaa/1364476010


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References

  • R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, NY, USA, 1963.
  • F. G. Boese, “Stability criteria for second-order dynamical systems involving several time delays,” SIAM Journal on Mathematical Analysis, vol. 26, no. 5, pp. 1306–1330, 1995.
  • Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, NY, USA, 1993.
  • S. Ruan and J. Wei, “On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 18, pp. 41–52, 2001.
  • S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 10, no. 6, pp. 863–874, 2003.
  • X. Li and J. Wei, “On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays,” Chaos, Solitons and Fractals, vol. 26, no. 2, pp. 519–526, 2005.
  • C. Huang, Y. He, L. Huang, and Y. Zhaohui, “Hopf bifurcation analysis of two neurons with three delays,” Nonlinear Analysis: Real World Applications, vol. 8, no. 3, pp. 903–921, 2007.
  • T. Zhang, H. Jiang, and Z. Teng, “On the distribution of the roots of a fifth degree exponential polynomial with application to a delayed neural network model,” Neurocomputing, vol. 72, pp. 1098–1104, 2009.
  • E. I. Jury, Inners and Stability of Dynamic Systems, John Wiley & Sons, 1974.
  • B. Zheng, L. Liang, and C. Zhang, “Extended Jury criterion,” Science China Mathematics, vol. 53, no. 4, pp. 1133–1150, 2010.
  • C. Zhang and B. Zheng, “Stability and bifurcation of a two-dimensional discrete neural network model with multi-delays,” Chaos, Solitons and Fractals, vol. 31, no. 5, pp. 1232–1242, 2007.