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Nonlinear Dynamics

, Volume 70, Issue 1, pp 475–479 | Cite as

Studying on the stability of fractional-order nonlinear system

  • Ling-Dong Zhao
  • Jian-Bing Hu
  • Jian-An Fang
  • Wen-Bing Zhang
Original Paper

Abstract

In this paper, we study the properties of the Mittag–Leffler function and propose an approach for calculating the maximum norm of eigenvalue of Jacobian matrix of nonlinear system. Then a simple approach is proposed for judging the stability of fractional nonlinear system. Based on the approach, the maximum decay rate of fractional nonlinear system can be calculated. Finally, some examples are provided to illustrate the approach.

Keywords

Fractional Mittag–Leffler Stability Decay rate Jacobian matrix 

References

  1. 1.
    Li, Z.B., Zhao, X.S.: The parametric synchronization scheme of chaotic system. Commun. Nonlinear Sci. Numer. Simul. 16, 2936–2940 (2011) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Li, Y., Chen, Y.Q., Podlubny, I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Krishna, B.T.: Studies on fractional order differentiators and integrators. A survey. Signal Process. 91, 386–426 (2011) MATHCrossRefGoogle Scholar
  4. 4.
    Jiang, X., Xu, M., Qi, H.: The fractional diffusion model with an absorption term and modified Fick’s law for non-local transport processes. Nonlinear Anal., Real World Appl. 10, 262–270 (2009) MathSciNetGoogle Scholar
  5. 5.
    Tang, Y., Wang, Z.D., Fang, J.A.: Pinning control of fractional-order weighted complex networks. Chaos 19, 013112 (2009). doi: 10.1063/1.3068350 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Shahiri, M., Ghaderi, R., Ranjbar, N., Hosseinnia, S., Momani, S.: Chaotic fractional-order Coullet system: synchronization and control approach. Commun. Nonlinear Sci. Numer. Simul. 15, 665–674 (2010) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Meral, F., Royston, T., Magin, R.: Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 15, 939–945 (2010) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ahmed, E., Elgazzar, A.S.: On fractional order differential equations model for nonlocal epidemics. Physica A 379, 607–614 (2007) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ahmad, W., El-Khazali, R.: Fractional-order dynamical models of love. Chaos Solitons Fractals 33, 1367–1375 (2007) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Song, L., Xu, S., Yang, J.: Dynamical models of happiness with fractional order. Commun. Nonlinear Sci. Numer. Simul. 15, 616–628 (2010) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Chen, D.Y., Liu, Y.X., Ma, X.Y.: Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn. 6(7), 893–901 (2012) CrossRefGoogle Scholar
  12. 12.
    Tang, Y., Fang, J.A.: Synchronization of N-coupled fractional-order chaotic systems with ring connection. Commun. Nonlinear Sci. Numer. Simul. 15, 401–412 (2010) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hu, J.B., Han, Y., Zhao, L.D.: Synchronizing fractional chaotic systems based on Lyapunov equation. Acta Phys. Sin. 57, 7522–7526 (2008) MathSciNetMATHGoogle Scholar
  14. 14.
    Sadati, S.J., Baleanu, D., Ranjbar, A., Ghaderi, R., Abdeljawad: Mittag–Leffler stability theorem for fractional nonlinear systems with delay. Abstr. Appl. Anal. 2010, 108651 (2010) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II. J. R. Aust. Soc. 13, 529–539 (1967) CrossRefGoogle Scholar
  16. 16.
    Daftardar-Gejji, V., Babakhani, A.: Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 293, 511–522 (2004) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Peng, G.J., Jiang, Y.L.: Two routes to chaos in the fractional Lorenz system with dimension continuously varying. Physica A 389, 4140–4148 (2010) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Ling-Dong Zhao
    • 1
    • 2
  • Jian-Bing Hu
    • 2
  • Jian-An Fang
    • 1
  • Wen-Bing Zhang
    • 1
  1. 1.College of Information Science and TechnologyDonghua UniversityShanghaiP.R. China
  2. 2.School of Electronics & InformationNantong UniversityNantongP.R. China

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