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Studying on the stability of fractional-order nonlinear system

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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.

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References

  1. Li, Z.B., Zhao, X.S.: The parametric synchronization scheme of chaotic system. Commun. Nonlinear Sci. Numer. Simul. 16, 2936–2940 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Li, Y., Chen, Y.Q., Podlubny, I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Krishna, B.T.: Studies on fractional order differentiators and integrators. A survey. Signal Process. 91, 386–426 (2011)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Meral, F., Royston, T., Magin, R.: Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 15, 939–945 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ahmed, E., Elgazzar, A.S.: On fractional order differential equations model for nonlocal epidemics. Physica A 379, 607–614 (2007)

    Article  MathSciNet  Google Scholar 

  9. Ahmad, W., El-Khazali, R.: Fractional-order dynamical models of love. Chaos Solitons Fractals 33, 1367–1375 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Song, L., Xu, S., Yang, J.: Dynamical models of happiness with fractional order. Commun. Nonlinear Sci. Numer. Simul. 15, 616–628 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hu, J.B., Han, Y., Zhao, L.D.: Synchronizing fractional chaotic systems based on Lyapunov equation. Acta Phys. Sin. 57, 7522–7526 (2008)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  15. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II. J. R. Aust. Soc. 13, 529–539 (1967)

    Article  Google Scholar 

  16. Daftardar-Gejji, V., Babakhani, A.: Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 293, 511–522 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Information Science and Technology, Donghua University, Shanghai, 201620, P.R. China

    Ling-Dong Zhao, Jian-An Fang & Wen-Bing Zhang

  2. School of Electronics & Information, Nantong University, Nantong, Jiangsu, 226019, P.R. China

    Ling-Dong Zhao & Jian-Bing Hu

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  1. Ling-Dong Zhao
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  2. Jian-Bing Hu
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  3. Jian-An Fang
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  4. Wen-Bing Zhang
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Correspondence to Jian-Bing Hu.

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Zhao, LD., Hu, JB., Fang, JA. et al. Studying on the stability of fractional-order nonlinear system. Nonlinear Dyn 70, 475–479 (2012). https://doi.org/10.1007/s11071-012-0469-0

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  • DOI: https://doi.org/10.1007/s11071-012-0469-0

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