Skip to main content
SpringerLink
Log in

Stability analysis of linear fractional differential system with multiple time delays

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

Abstract

In this paper, we study the stability of n-dimensional linear fractional differential equation with time delays, where the delay matrix is defined in (R +)n×n. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. As its an application, we apply our theorem to the delayed system in one spatial dimension studied by Chen and Moore [Nonlinear Dynamics 29, 2002, 191] and determine the asymptotically stable region of the system. We also deal with synchronization between the coupled Duffing oscillators with time delays by the linear feedback control method and the aid of our theorem, where the domain of the control-synchronization parameters is determined.

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. Hale, J.K., Verdugn Lunel, S.M., Introduction to Functional Differential Equations. Springer-Verlag, New York (1991)

    Google Scholar 

  2. Deng, W.H., Wu, Y.J., Li, C.P.: Stability analysis of differential equations with time-dependent delay. Int. J. Bif. Chaos. 16(2), 465–472 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Li, C.P., Sun, W.G., Kurths, J.: Synchronization of complex dynamical networks with time delays. Physica A 361, 24–34 (2006)

    Article  Google Scholar 

  4. Li, C.P., Sun, W.G., Xu, D.: Synchronization of complex dynamical networks with nonlinear inner-coupling functions and time delays. Prog. Theor. Phys. 114(4), 749–761 (2005)

    Article  MATH  Google Scholar 

  5. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience, New York (1993)

    MATH  Google Scholar 

  6. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

  7. Butzer, P.L., Westphal, U.: An Introduction to Fractional Calculus. World Scientific, Singapore (2000)

    Google Scholar 

  8. Agrawal, O.P., Tenreiro Machado, J.A., Sabatier, J.: Introduction. Nonlinear Dyn. 38(1–2), 1–2 (2004)

    MathSciNet  Google Scholar 

  9. Kempfle, S., Schäfer, I., Beyer, H.: Fractional calculus: theory and applications. Nonlinear Dyn. 29(1–4), 99–127 (2002)

    Google Scholar 

  10. Matignon, D.: Stability result on fractional differential equations with applications to control processing. In: Proceedings of IMACS-SMC, pp. 963–968, Lille, France (1996)

  11. Chen, Y., Moore, K.L.: Analytical stability bound for a class of delayed fractional-order dynamic systems. Nonlinear Dyn. 29, 191–200 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Muth, E.J.: Transform Methods with Applications to Engineering and Operations Research. Prentice-Hall, New Jersey (1977)

    MATH  Google Scholar 

  13. Franklin, P.: Functions of Complex Variables. Prentice-Hall, New Jersey (1958)

    MATH  Google Scholar 

  14. Chen, Y., Moore, K.L.: Analytical stability bound for delayed second-order systems with repeating poles using Lambert function ω. Automatica 38, 891–895 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Deng, W.H., Li, C.P.: Synchronization of chaotic fractional Chen system. J. Phys. Soc. Jpn. 74, 1645–1648 (2005)

    Article  MATH  Google Scholar 

  16. Deng, W.H., Li, C.P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61–72 (2005)

    Article  Google Scholar 

  17. Lu, J.G.: Nonlinear observer design to synchronize fractional-order chaotic system via a scaler transmitted signal. Physica A 359, 107–118 (2005)

    Article  Google Scholar 

  18. Lu, J.G.: Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. Chaos Solitons Fractals 27(2), 519–525 (2006)

    Article  MATH  Google Scholar 

  19. Zhou, T.S., Li, C.P.: Synchronization in fractional-order differential systems. Physica D 212, 111–125 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Li, C.P., Deng, W.H., Xu, D.: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171–185 (2006)

    Article  MathSciNet  Google Scholar 

  21. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002)

    MathSciNet  Google Scholar 

  22. Li, C.P., Peng, G.J.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22, 443–450 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Weihua Deng

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Weihua Deng & Changpin Li

  3. Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, China

    Jinhu Lü

Authors
  1. Weihua Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changpin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinhu Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changpin Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deng, W., Li, C. & Lü, J. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn 48, 409–416 (2007). https://doi.org/10.1007/s11071-006-9094-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-006-9094-0

Keywords

Search

Navigation