Skip to main content
SpringerLink
Log in

Exponential stability and spectral analysis of a delayed ring neural network with a small-world connection

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper addresses the dynamical behavior of the linearized delayed ring neural network system with a small-world connection. The semigroup approach is adopted in investigation. The asymptotic eigenvalues of the system are presented. It shows that the spectrum of the system is located in the left half complex plane and its real part goes to −∞ when the connection weights between neurons are well-defined. The spectrum determined growth condition is held true and the exponential stability of the system is then established. Moreover, we present the necessary conditions for the neuron and feedback gains, for which the closed-loop system is delay-independent exponentially stable, and we further provide the sufficient and necessary conditions when the concrete number of neurons and the location of small-world connection are given. Finally, numerical simulations are presented to illustrate the convergence of the state for the system and demonstrate the effect of the feedback gain on stability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bélair, J.: Stability in a model of a delayed neural network. J. Dyn. Differ. Equ. 5, 607–623 (1993)

    Article  MATH  Google Scholar 

  2. Hagen, T.: Asymptotic solutions of characteristic equations. Nonlinear Anal., Real World Appl. 6(3), 429–446 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. 81, 3088–3092 (1984)

    Article  Google Scholar 

  4. Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Feedback Time Delays. Springer, Heidelberg (2002)

    Google Scholar 

  5. Jury, E.I.: Inners and Stability of Dynamic Systems, 2rd edn. Wiley, New York (1982)

    MATH  Google Scholar 

  6. Langer, R.E.: On the zeros of exponential sum and integrals. Bull. Am. Math. Soc. 7, 213–239 (1931)

    Article  Google Scholar 

  7. Lasalle, J.P.: The Stability and Control of Discrete Processes. Springer, New York (1986)

    Book  MATH  Google Scholar 

  8. Li, C., Chen, G.: Local stability and Hopf bifurcation in small-world delayed networks. Chaos Solitons Fractals 20, 353–361 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, C., Chen, G.: Stability of a neural network model with small-world connections. Phys. Rev. E 68, 052901 (2003)

    Article  Google Scholar 

  10. Luo, Z.H., Guo, B.Z., Morgul, O.: Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer, London (1999)

    Book  MATH  Google Scholar 

  11. Marcus, C.M., Westervelt, R.M.: Stability of analog neural networks with delay. Phys. Rev. A 39, 347–359 (1989)

    Article  MathSciNet  Google Scholar 

  12. Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  Google Scholar 

  13. Shkalikov, A.A.: Boundary value problems for ordinary differential equations with a parameter in the boundary conditions. J. Sov. Math. 33, 1311–1342 (1986)

    Article  MATH  Google Scholar 

  14. Wang, J.M., Guo, B.Z., Fu, M.Y.: Dynamic behavior of a heat equation with memory. Math. Methods Appl. Sci. 32(10), 1287–1310 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, J.M., Lv, X.W., Zhao, D.X.: Exponential stability and spectral analysis of the pendulum system under position and delayed position feedbacks. Int. J. Control. 84(5), 904–915 (2011)

    Article  Google Scholar 

  16. Wang, L., Xu, G.Q.: Spectral analysis and expansion of solution to a class of delay differential equations. Math. Sci. 29A(4), 843–857 (2009)

    Google Scholar 

  17. Watts, D.J., Strogatz, S.H.: Collective dynamics of “small world” networks. Nature 393, 440–442 (1998)

    Article  Google Scholar 

  18. Xu, X.: Complicated dynamics of a ring neural network with time delays. J. Phys. A: Math. Theory 41, 035102 (2008)

    Article  Google Scholar 

  19. Xu, X., Wang, Z.H.: Effects of small world connection on the dynamics of a delayed ring network. Nonlinear Dyn. 56, 127–144 (2009)

    Article  MATH  Google Scholar 

  20. Xu, X., Hu, H.Y., Wang, H.L.: Dynamics of a two dimensional delayed small-world network under delayed feedback control. Int. J. Bifurc. Chaos 16, 3257–3273 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Xu, G.Q., Yung, S.P.: Properties of a class of C 0 semigroups on Banach spaces and their applications. J. Math. Anal. Appl. 328, 245–256 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, PR China

    Dong-Xia Zhao & Jun-Min Wang

  2. Department of Mathematics, North University of China, Taiyuan, 030051, PR China

    Dong-Xia Zhao

Authors
  1. Dong-Xia Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun-Min Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dong-Xia Zhao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, DX., Wang, JM. Exponential stability and spectral analysis of a delayed ring neural network with a small-world connection. Nonlinear Dyn 68, 77–93 (2012). https://doi.org/10.1007/s11071-011-0205-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0205-1

Keywords

Search

Navigation